A team of astrophysicists led by scientists from Texas and California has now been able to observe the most distant galaxy ever seen: approximately 13.1 billion light years
Much larger than the Milky Way, and produces stars at a tremendous rate. Imaging of galaxy z8_GND_5296, courtesy of STScI/NASA
Our universe was created in the Big Bang about 13.8 billion years ago, and since then it has been expanding at an ever-increasing rate. Therefore, the more distant galaxies we observe from us, the more ancient they are. A team of astrophysicists led by scientists from Texas and California has now been able to observe the most distant galaxy ever seen: approximately 13.1 billion light years. This means that light took 13.1 billion years to reach us, and we see it as it was only 700 million years after the Big Bang - literally the birthplace of the universe. "We see a very young galaxy, at a very early stage of the development of objects in the universe," says the Israeli partner in the study, Prof. Avishai Dekel from the Hebrew University in Jerusalem. "It is a very large galaxy, producing stars at a rate that is 100 or 200 times more than our galaxy today. This teaches about the very, very early stages of the development of those galaxies that we live in today, and where stars and planets and life like us were formed. These are our ancient parents."
far red
The galaxy, known in the meantime as z8_GND_5296, was first discovered in the observations of the "Hubble" space telescope, as part of a project to scan the sky and search for extremely distant star systems. To verify the find, the researchers examined it in the largest telescope in the world, the "Kek" observatory in Hawaii, where the galaxy can also be photographed with a spectroscope - a device that analyzes the composition of the light emitted from it. When you see that the light of different elements like hydrogen for example, is much redder than it should be, it means that the galaxy is moving away from us quickly. This method is similar to the Doppler phenomenon, in which we hear a decrease or increase in the frequency of sound waves, if they are approaching us or moving away from us. For example - when an ambulance is speeding towards us, the waves sound denser, but when it moves away, the frequency of the waves reaching us decreases. The same phenomenon occurs in light waves, when it comes to much higher speeds. A celestial object approaching us at enormous speed will appear bluer to us, and if it moves away quickly, we will see the phenomenon known as "red shift". In the current galaxy a higher redshift was measured than any other galaxy (7.5 compared to the previous record which was 7.2), which makes it possible to calculate its speed relative to us, and thus determine its distance.
The beginning of the universe
The scientists today know how to describe with a good degree of accuracy (at least until it is proven otherwise) the sequence of events in the first units of time after the big bang. The enormous energy that was released turned into matter, and gained mass (partly thanks to the Higgs boson, whose existence was almost definitively proven only recently). Those particles combined to form protons and neutrons, which with the electrons formed the smallest atoms, hydrogen and helium. The first atoms clustered into stars (suns) where conditions of heat and pressure were created that allow fusion into even heavier atoms, and in their explosions, really heavy atoms, such as metals, were also created. Remnants of such explosions and dust particles coalesced to form planets, moons and other celestial bodies. However, scientists still know relatively little about the development processes of the universe after the first period: how the galaxies were formed, how they were dispersed as they were, how the conditions for star formation were created inside, and so on. Our improved ability in recent decades to observe the earliest galaxies may improve our understanding of how these processes occurred and how our galaxy was ultimately formed.
269 תגובות
The issue of quantum space came up here.
As a reminder - I do not at all accept the assumption that this type of space-time derives from the story of Achilles and the tortoise, but I would like to suggest that we all think about how this type of space-time is supposed to deal with the physical phenomena known to us. I only propose to think - I don't think that with the current state of knowledge of anyone in the world it is even possible to discuss these questions seriously - and therefore I only propose to think and not start a discussion.
for example:
1. According to the assumptions of the theory of relativity, it is impossible to know if a body moves between one space quantum and another because there is nothing that deserves to be called absolute motion/rest. This is not a question of measurement accuracy because it is also impossible to know whether a body has passed through a trillion trillion trillion trillion trillion quanta of space or if none have passed through.
2. How can inertial motion in a straight line be explained? After all, in quantum space a body must (in most directions) move in some sort of zigzag, so how can it "remember" the direction from zig to zig? Regarding macroscopic bodies, it can be said that the "zigzags" of the various particles are offset, but regarding microscopic bodies, this cannot be claimed. Besides - there is also a problem with macroscopic bodies because this means that a body in motion has to heat up (due to the relative motion imposed on its particles) and this should be possible to notice and this - not only does it not agree with the theory of relativity - it is also not a phenomenon that anyone has noticed.
3. I am sure that anyone who thinks about the subject will find a few more goats. It may be possible to model everything correctly but personally, at this stage, I prefer to let Ockham decide.
Miracles:
Good. One last time really because there is a limit to every waste of time:
The solution to the paradox is indeed mathematical and that is because the paradox is mathematical. It has nothing to do with the physical world! nothing!
The one who makes the connection between them is you and you alone.
The mathematical world is part of the philosophical world, but paradox is in the mathematical part (assuming - and this is an insignificant matter of defining words only - that logic is part of mathematics).
Russell's paradox or any other paradox is something that turns out to be an apparent logical contradiction between two apparently legitimate ways of drawing conclusions.
It's part of the logic. Zeno's paradox is no exception to this rule.
There are no known paradoxes in mathematics.
Russell paradox solved. In fact all the paradoxes that have ever arisen have been resolved.
Some of them were solved by new insights in mathematics itself (identifying an error in the rules of definition and drawing conclusions in mathematics) and some were solved by identifying an error of those who formulated them.
When they formulated the non-Euclidean geometries, it was not because Euclid discovered a mistake about the nature of the physical world.
They understood that it was possible to formulate other axioms and get a different geometry, but they didn't know what was true in our world.
In fact, Euclid's "mistake" (if there was one at all - I don't know if anyone wrote what went through Euclid's mind in this regard) was (again - maybe - it is not known if there was one at all) assuming that the physical world corresponds to Euclidean geometry. This is a mistake in the field of physics. In the field of mathematics he had no mistake. Obviously he knew that if he formulated other axioms something different would be obtained, but he simply did not think there was any point in formulating other axioms because it would not be relevant to our world (this is a physical thought).
Euclid's postulates were first proven wrong (again! physically!) in 1919 in Eddington's experiment.
Zenon's conclusion was:
Achilles will "never" catch the tortoise. That is, there is no point in time in the future when Achilles will get the tortoise.
And the correct wording is:
The point where Achilles catches the tortoise will never belong to the set of points described in the problem.
And of course it must be remembered that this is not about the movement of any physical body. Achilles described in the problem is not a physical body but a mathematical point. No physical properties of Achilles are mentioned here. Only a correspondence between the natural numbers and three series of points is mentioned:
a) A series of time points.
b) A series of distance points describing Achilles' position.
c) A series of distance points describing the location of the turtle.
The time segment described by Zeno is a blocked segment.
The claim "never" cannot exist when talking about matching in a blocked section of time.
Since the described "Achilles" is a point body, there is no meaning to questions such as:
a) In which hand.
b) In which leg.
Since the section is blocked, the question has no meaning:
c) Who will pass the baton.
Within the blocked section Achilles will not reach the tortoise. He will catch it at a point that is outside the section.
I wrote a comment yesterday that disappeared...
Michael - you wrote that the solution to Achilles' paradox is with the help of the column limit. You wrote that I also use a mathematical model to explain nature. Now it seems to me that you are twisting it... (Probably my misunderstanding).
I didn't mention math. Zeno's paradoxes do not belong to the world of mathematics. They belong to philosophy.
And Michael - there are many paradoxes in mathematics, it just depends on your definition of a paradox. Think Russell, Curry, Tarski and Banach, and Gedel, for example.
Euclid's error was discovered 100 years before Einstein - and completely by chance - precisely with the help of a thought experiment!
Israel - I say again and again and again: Zenon did not talk about a problem at the border of the column. He was talking (to my understanding) about the infinity of the series. There is a fundamental difference here.
MouthHole
I don't believe there is an ego issue here.
Nissim claims that Zeno's paradox is unsolved in philosophy. He also claims that mathematical proofs are not necessarily acceptable in the world of physics. If you followed the details of the discussion, the way I tried to show him that in this particular case the paradox is actually quite resolved, is similar to the way you find out if the sum of an infinite column converges or diverges.
For example: How will you know if the sum of the column ....1 +3/4 -5/6 +1/2-7/8 converges to a finite value or diverges to infinity?
The way to find out is to look at the small details, group similar groups and compare them. This is what I tried to do here, and therefore the descent into the details of the details. I hope my friend Nissim took it that way.
Prove I'm right? I don't have much interest in psychology, although technically the opposite is right and wrong.
Machal, I accept that I was wrong. I was of course just guessing (and I only guessed that from the way I behaved in the past).
In any case, I don't think anyone who has read the dance of comments here will think so. So I guess you cater to the lazy readers who only read the last few comments. Really nice of you 🙂
Israel
You are really wrong. Zenon was not interested in the sum of the distances. He talked about the meaning of infinite events. And this is a problem that has not been solved to this day, according to a number of philosophers. And, as strange as it is, in computer science.
Michael
Euclid's mistake was discovered about 100 years before Einstein. And the error in the interpretation was discovered by a thought experiment "let's assume that the axiom of parallels is not true and then we will find a contradiction".
Are there paradoxes in mathematics? First you need to define a paradox. Any paradox I describe - you will say it is not a paradox and it becomes an argument about semantics. I'm a Russell fan, and for some reason there's a math paradox named after him. And there are paradoxes named after people like Carey, von Neumann, Banach and Tarsky, and of course Gadel.
And Michael, for some reason you are twisting my words. I was not talking about a paradox in mathematics. I was talking about a paradox in physics. I do not claim that Zeno's paradoxes require that the world is discrete. But it is thought provoking. At least for me.
Mouth Hole:
I think your interpretation is not accurate.
I know I'm right even without reacting and I enjoy it just as much.
The reason I still respond from time to time is so that I don't get the impression that I got the wrong opinion.
There are, as mentioned, many errors here and the last one that came up here again is the claim that reverses the roles and presents Nissim as the one who supports the position that mathematics should not be imposed on physics when the situation is exactly the opposite, as I explained once again.
I see this discussion like this: someone with a huge ego, even though inside he realized he was wrong, is not ready to admit it. And two more people who enjoy being right and explain the same thing to him over and over again.
I am not blaming anyone here, what I wrote sounds sarcastic because I do see something negative in the discussion here and I would like it to change (of course you don't have to agree with me, this is just my opinion): at some point, once all the things have been said, there is no need to repeat Nissim is allowed to have the last word, even if he is wrong. This is how we will save a carousel...
Sometimes I also have a problem understanding that I'm wrong, and I need something from the outside to tell me to let go so that I can really think about it again (and not think of a way to make myself right).
And sometimes I really feel like being right no matter how many times I repeat myself and I need someone to tell me to let go in order to understand that I am repeating myself again and again for no unnecessary reason.
So let go. 🙂
In general, I think that most of the real debates and claps I participate in are the ones where I'm not 100% sure of my claim and then realize I'm wrong. That's actually how you both learn new things and pay attention to it (which is also quite a difficult action for the most part)
For those who still don't understand:
Euclid's "mistake" was not discovered through a thought experiment but thanks to physical findings.
Without physical findings that confirm Einstein's model and reject Euclid's model, no one would conclude that Euclid was wrong. On the contrary: they would decide that Einstein was wrong.
There is no one in the history of science who has projected something onto the physical world from a paradox in mathematics (partly because paradoxes in mathematics are unknown).
And yet - this is what Nissim is trying to do.
The point is that there is not even a paradox in mathematics and all that is here is Zeno's ignorance and therefore Nissim casts miracles on physics from the reflections of his heart and nothing else.
I agreed with you in the past that the projection from mathematics to physics is not always completely clear, because of an infinitesimal that can be as small as we want in mathematics but has a finite size in physics.
This is not the case in Zeno's paradox.
Because according to the original Zeno, Achilles will not reach the tortoise because the distance he has to travel to do so, i.e. 100 meters +10+1... is the sum of an infinite number of distances, small and decreasing indeed but still an infinite number, and this distance must therefore be greater than 112 meter.
This is a fairly reasonable assumption 3000 years ago, but not today. Today we know that mathematically, and also physically, the distance is less than 112 meters.
And so the paradox is resolved.
Israel
Dismissing the paradox because of some equation that calculates the limit of a column is irrelevant, in my opinion. You are missing out on a lot of great ideas. Just an example: if Achilles checks at each step if the number of steps he has taken is the sum of two primes, then by the time he reaches the turtle he will prove (or disprove) the Goldbach hypothesis. He will of course also solve the stopping problem, and so on....
Beyond that, what bothers me is that they throw from mathematics to the physical world, without having a solid basis to think that such a projection is indeed valid. Mathematics is based on axioms. It is not an absolute system. Euclid was wrong about this in the field of geometry. It's a shame to repeat this mistake again 🙂
What happened to you, our brother, in poison?
What is all this emotion, you are no longer allowed to write short poems on this site?!
And we don't think we're that smart at all.
Proud..
Have a good day.
I don't think you are as smart as you think.
Your'e right. Really cool.
Because otherwise there would be no balance
Come in!
All the columns here gathered!
I sat on Mount Olympus,
Paradox about Achilles I wrote,
Suddenly the column gathered,
The meeting will not be seen as a conference again.
Zenon's paradox is not stupid at all, but stems from a lack of knowledge about the final limit of certain convergent series, knowledge that did not exist in Zenon's time.
Miracles:
I don't agree with anything you said in your last comment but I won't spend any more time on it
Michael
So at least you agree that there is no connection between the paradox (allegedly...) and the limit of a geometric column. Right?
To say that the whole paradox idea is stupid... It could be true 🙂 I understand that this is also the essence of Israel's response (just like the idea of saying that the border of Tor is a solution to the paradox is stupid).
Let me think about it for a night 🙂
I sat on Mount Olympus
And I wrote a paradox about a turtle
Suddenly the column came together
And this is the end of the paradox.
Ruth end.
Miracles
The column limit you displayed is lower than 112.
No matter how many members we add to the column, at infinity or at the end, we will not reach 112 in the end.
And according to Zenon, we will reach and pass 112.
And here is his mistake, and also the end of the paradox.
Miracles:
Right. But the question is just like the one you ask. You ask - out of the collection of Achilles that you have built in such a way that none of them will reach the tortoise - which Achilles will reach the tortoise.
This is just as stupid a question.
Israel
I do not understand what you are saying. The limit of the column is...111.11111111111111111111
What does 112 have to do with this?
I mention infinity because this is the essence of the problem... an infinity of events.
Michael
If they don't move then Achilles won't reach the tortoise.
Miracles:
No. I'm showing you how ridiculous your line of thought is and you don't see the ridiculousness in it.
Knows what? I will demonstrate it to you in your favorite model. That of quantum space: space is quantum and Achilles is at point A.
The turtle is at another point - B.
Achilles did not move from point A and the tortoise did not move from point B.
Which leg of Achilles will be in front when he reaches the tortoise?
Miracles
From my response:
"So I repeat and ask: Do you accept that according to Zenon's argument 100 meters + 10 meters + a meter + one tenth of a meter... up to any number of intervals we want (smaller than infinity, even by a lot), is greater than 112?"
So why do you keep mentioning infinity? Are you talking about another paradox?
Yossi Simon
You invented a strange definition for a paradox and claimed that Zeno's paradox does not fit your definition.
Michael
Are you trying to convince me I'm right?
Israel
For the 100th time plus 10 plus 1 plus 0.1 ….. I know how to summarize a geometric series. The problem is not the amount. The problem is that there are endless events. There is no relation to the time between the events (this time constitutes a converging column).
To try to explain the problem - I added the matter of "moving the stick". Now it should be clear that there is no one to pass the baton to that runner who is on the equinox line.
Yossi Simon:
You use the correct proof technique but from the wording of the explanation it becomes clear that you do not understand it.
I'll give you a chance to correct the wording before I explain the mistake.
On the fact that your approach to Zenon's paradox is nothing but evasion, I do not find it appropriate to expand at all because we have already corresponded on the matter.
Miracles:
So please tell me, if there is a group of Achilles who all wear a black hat, what is the name of the mother of Achilles (from the group) who wears a white hat.
Sabdarmish Yehuda!
Below is the mathematical proof that the power of the set of real numbers is greater than the power of the set of rational numbers.
1. The set of rational numbers is a set of numbers with a divisor.
2. We will construct a number and prove that it does not belong to the group of rational numbers and therefore this group is not rational.
3. Any number can be represented by infinitely many digits.
4. Now we will determine the digits using the following method. If the first digit in the first number is 3, we will place the value 3 in the first digit, otherwise we will place the digit 4.
5. And now we will move to the second digit and in the same way we will place 3 or 4 depending on the value of the second digit in the second number.
6. We received a number that is not included in the rational numbers.
And again Zenon's entire paradox rests on only one claim A leads to B (and since the claim is fundamentally unfounded there is no need to clarify anything) similar to someone who proves that the sum of the angles in a triangle is 170 degrees and relies on the theorem that the triangle is 180 degrees
Miracles
Zeno's entire argument is that the sum of all infinitely spaced intervals equals infinity as well.
This is true in many cases but not in this particular case. Not even in amounts smaller than infinity. This is the Achilles heel of the argument.
So I repeat and ask: Do you accept that according to Zenon's argument 100 meters + 10 meters + a meter + a tenth of a meter ... up to any number of intervals we want (smaller than infinity, even by a lot), is greater than 112?
Do you accept that this is the original paradox?
And do you accept that this is simply not true? That is, any such sum will always be less than 112?
If you did, then the original paradox is solved.
What I understand from the comments is that no one has an answer to the question. That's what I've been trying to say all along.
Michael
My response is completely coherent. Let's leave it at that.
ב
I flow with you. You claim that there is no right and no left in mathematics. I will not try to argue with this strange logic.
Miracles:
The paradox is about a mathematical point called Achilles.
It's not a person.
Nor is it any other physical body.
This is a mathematical point.
If a mathematical point is named after Achilles
So this point has neither a right hand nor a left hand. Neither the right leg nor the left leg.
Miracles:
This is already a completely pointless response.
I have no idea what you want.
If we go back to the previous responses, which were at least syntactically correct (even though they were planned incorrectly), then if we summarize your question, it can be summarized as follows:
A list of Achilles is given, none of which reaches 1024. Find Achilles' trait X that reaches 1024?
Or in a parable: given a list of Achilles who all have a black hat. What color are the eyes of Achilles with the red hat?
Michael
Let's say that Achilles runs 512 meters per minute, so you must be talking about the same Achilles that arrives in exactly 2 minutes. We will call him Achilles A-0. Or maybe Achilles A-1?
Miracles:
Of course, what I'm missing is a definition of Achilles that reaches a distance of 1024.
Michael
I thought you had a formula for that too…. 🙂
What detail are you missing to answer me? What are you missing in the problem description? If it is important to you - assume that the first Achilles starts with the right hand.
Miracles:
Maybe you will ask me now how tall Achilles is named Joseph?
Miracles:
It's good that you don't ask me what color Achilles' hair will be when he reaches the 1024 line or what his mother's name is or what his shoe number is.
How many times do you have to say that it is impossible to answer a question you ask about a situation that you did not address in the description of the problem?!
I can't say anything about Achilles reaching line 1024 because no Achilles in the series you described would reach it.
Michael
Yes. But you can't tell me that that Achilles at the 1024 meter line will get the stick.
Let's go back to Thomson's lamp, and suppose that each Achilles changes hand. In which hand will Achilles receive the stick at the 1024 meter line?
By the way - I think you also can't tell me what time Achilles at the crossing line will receive the stick - because that entails measuring time with infinite precision.
Suppose that the Michaelson-Morlay experiment was carried out in intergalactic space over large enough distances.
If the intergalactic space is different from the space inside a galaxy in that it "inflates" then the results of the experiment should also be different from the results of the same experiment inside a galaxy.
Miracles:
That is - the question illustrates a problem if it pretends to talk about what it is not talking about.
If you ask what will happen in this situation without pretending to talk about what will happen outside the scope of the definition then it is very simple:
By the time it takes a single Achilles to run the entire distance of 1024 meters, all the infinite Achilles in the question will have run and for each one of them I can tell you when he will start his run and when he will finish it
Miracles:
Indeed the question illustrates a problem... in the question itself.
That's why I'm wondering what you're asking
Michael
The question tries to show a problem. If every Achilles passes the stick to the next Achilles, who is halfway to the meeting line, then there is no Achilles to pass the stick to the Achilles who is at the 1024 meter line.
Miracles:
I'm not rowing anywhere.
You asked a question and I'm trying to understand what you're asking.
So I have no idea where you are headed.
Miracles:
I already guessed that this was the intention and answered it
Michael
He starts at the starting line.
At point A.
1024 meters from the meeting line.
He will run 512 meters and pass the baton to Achilles-1.
Miracles:
It seems to me that the wording of the question misled me because I thought you were talking about distances from the start of the route.
If you are talking about distances from the end of the track (ignoring the fact that the end moves away with the movement of the turtle, but that is not what is important) then you have not defined what happens from a distance of 1024 onwards (and in fact you have not defined how to get to 1024 either) and it is not clear to me how you expect me to answer the question.
Miracles:
You did not understand.
I didn't ask what his name was.
I asked who he is - where is he standing?
We will call the first runner Achilles-0, the second (the one you meet at 512 meters) Achilles-1 and so on. Judah will tell you (perhaps) that the "last" runner is Achilles-a-0...
Miracles:
If everything is set then tell me who is the first messenger.
Contrary to Zenon's question - here it is important to know who the first messenger is because he is supposed to be the one to start the race.
In Zenon's question, the identification of the last event in the series is not important (and indeed there is no last event) because these events are passive. They are just a description of a situation that happens even without being addressed.
B:
This is exactly the case with the Big Bang.
No one pretends to understand exactly what happened in his first fractions of a second or "before him".
The word "before him" is in quotation marks because there is a question about the meaning of the word in the absence of time-space, but we simply do not know.
There are quite a few speculations about it, but there are no answers yet.
A. Zero is the infinity of the natural or rational numbers. Only A is the infinity of real numbers. It can be proven that any match you take that compares the natural and the real will not contain all the real. Therefore A is greater than A zero.
This is what I remember from Cantor's set theory
good week
Sabdarmish Yehuda
Michael
I don't see what is not defined in the problem I proposed. Achilles starts at point A, and the tortoise at point B. They are supposed to meet at a distance of 1024 meters from A.
Michael:
I think we agree.
We both agree that:
a) "In the series described by Zeno, Achilles will not catch the tortoise".
b) Achilles will indeed get the tortoise in the end and this will not happen in the realm of existence of the series.
a question:
Is it not possible that there is a similar problem in the field of drawing conclusions about the existence of the Big Bang?
That is, isn't it possible that there is a problem of a wrong description of the occurrence in nature? A description that is good for a certain field but cannot exist outside of that field? Or a description that is accurate in a certain area but the accuracy decreases as you move away from that area?
B:
I see I answered a question you didn't ask.
That is - some of the things do refer to the question you asked, but the emphasis is elsewhere.
It is clear that at points in the series described by Zeno, Achilles will not reach the tortoise.
B:
If this is the meaning then surely it does not follow that at other points Achilles will not get the tortoise.
This was exactly Zenon's mistake - he tried to deduce from what is happening in a blocked time section what is happening along the entire timeline.
In the blocked time section Achilles really does not catch the tortoise but after it he does catch it.
Michael:
In the example you provided, the problem data are:
The first time Achilles advances 10 meters.
The second time Achilles advances 1 meter.
And so on.
There is a point in time up to which Achilles maintains these data.
However, from there on Achilles continues to advance but does not maintain these data.
B:
I don't know what the phrase "maintains the data of the problem" means.
Michael:
A) Thanks for the explanations.
b) In your example you meant that Achilles does 10 meters and then 1 meter and then the tenth of a meter and so on.
I did not understand the example because of a certain reluctance in its wording.
c) You are right. There is a point in time from which onwards the conduct according to the data of the problem is impossible. This happens even if this point in time is exactly the point when Achilles gets the tortoise and even if that point is before the time when Achilles gets the tortoise.
That is, the very presentation of Achilles' movement in the way described in the problem is good up to a certain point in time and no later than that.
That is, there is a point in time from which Achilles continues to move, but his movement is not as described in the problem.
d) But doesn't it imply that whenever Achilles fulfills the data of the problem he will not get the tortoise?
sympathetic:
There is a limit to every trick.
I won't start arguing with you about the meaning of the word gossip.
I have more important things in my life.
In my opinion, this is a disrespectful statement that you said, the whole purpose of which is defamation, but I will not spend any more time on it.
B:
You're just missing the point.
Zeno's mistake is in establishing a conclusion based on what will happen outside the time frame in which the events he described occur.
I showed you a variation on the same mistake when the limit of the time series is not at all at the time of the encounter.
I took a certain speed because this is the speed we talked about all the time (and I remind you thatThe first attempt in this discussion is to present an answer to the paradox done by while my addressing you toAn article I wrote on the subject In addition to the fact that I mentioned the speed, I also made it clear in a way that is not ambiguous that I am talking about the time series) but for any pair of speeds that you give in relation to Achilles and the turtle, I can do exactly the same exercise.
Therefore, the consideration you tried to point to as the decisive consideration - that the limit of the series is not included in the series - is meaningless.
The only significant thing is that Zenvan draws conclusions about the entire time span from a series of events that is bounded in time (blocked).
By the way: the ways in which people deceive themselves arouses real wonder in my eyes:
My response that I pointed to was to your response - one before it - in which you brought the paradox as evidence that mathematics does not fully describe reality.
At this point you did not understand the matter at all.
You began to understand it only after my many explanations and now you see fit to teach me about it.
Michael:
Sorry. In the paradox I was referring to there is a given:
1) Achilles is faster than the tortoise but the tortoise is ahead of Achilles.
2) Until Achilles reaches the point where the turtle is standing meanwhile the turtle has moved to another point and this happens at every stage.
Someone on Wikipedia decided to set the initial distance as 100 meters, Achilles' speed as 10 meters per second, and the turtle's speed as 1 meter per second.
And so you can really set the times in seconds.
But the essence of the problem is not different.
From the data of the problem it is possible to deduce the existence of the limit of the time series. But the data does not talk about the location of Achilles in the border. We get the position of Achilles at the limit of the time series only from the knowledge that his movement is continuous and not from the descriptions of the position included in the formulation of the problem.
From_Yachal
This is nonsense. Scientific information is provided in relevant units and sizes. to claim to be discovered
A particle is needed with the highest certainty at least above 3 sigma. A claim that a particle has been discovered
50% or 75% is not a scientific claim, it is nonsense intended for publication. For your information in the field
of particle research it is much easier to claim that a particle has not been found than that it has been found, thus
For researchers, the time required to prove that a particle has not been found is much shorter than required
prove that it is found. When there are single events outside the noise they should be analyzed carefully
and assess the statistical chance of receiving such an event only after all options have been examined
You can make an announcement. A scientific claim that a particle is found with a probability of 50% is valid
For other "scientific" claims such as for example that there was life on Mars, etc... which claims to be possible
To claim with 50% or 75% certainty about the existence of a particle does not understand what science is and how it is
Works, but the scientist deals with questionable publications if they fit his line of thinking
and does not publish scientific news that allegedly points to problems in the existing theories.
The fact that you think there is no good explanation for dark matter observations still does not make your theory
The dark matter to be true only points to the limitations of the researchers and their theories in the field. any experiment
His goal of finding dark matter and failing only challenges the dark matter theory, but the scientist does not
Advertiser... The foolish followers of course claim that an experiment that does not detect a particle has no scientific significance
Because he did not contradict the theory (it is always possible to invent some other particle, with a greater mass
or other mysterious properties).
By the way, if the measurement in the previous news was correct, what happened to it during the past four years??
After all, it was clear that this was news aimed at obtaining additional funding, so they turned to the general public.
By the way, in:
בDescription of the problem on Wikipedia - the one we talked about all the time - explicit speeds are also taken.
And for the sake of historical accuracy - we have never argued about whether dark matter particles have been discovered, but only about whether there is dark matter and whether there is a better explanation of it for the observations.
incidentally:
Of course, the mere advertising of 50% is an advertising of doubt.
Even innews about 75% Doubt is expressed.
Your claim against the knowledge site is simply not justified.
sympathetic:
I remember something about news in which it was claimed that it was discovered with a high probability.
I do not accept the claim that it was complete nonsense (on the contrary: in my opinion, to claim based on the fact that the current detectors have not yet discovered dark matter that that news was complete nonsense - is complete nonsense).
First of all: we all know what probability is and when it is not 100% (and it was much less than XNUMX% in the above case) then there is also the possibility of a wrong result.
Secondly (and here this requires clarification) it could be that the method of measurement in that information was different and that the current non-disclosure does not contradict it at all.
B:
Unbelievable!
It is clear that the column I am talking about is a column of times! This is what I do all the time and my whole point is that you cannot draw a conclusion about "ever" from a blocked time series!
When people talk about explicit times (and we are talking about explicit times which I also mentioned to show that even you assumed a known speed and therefore the numbers 10, 11, 11.1, 11.11 etc. never bothered you) this means that they adopted (for the sake of illustration) the speed of 10 meters per second for Achilles .
In general - when you express yourself in the form of "until X happens, Y will also happen" it is clear that you are talking about times.
In short - if you didn't understand what I keep talking about, then I ask you to understand it now: the column I'm talking about is a column of times.
Zeno also talked about times because the phrase "Achilles will never catch the tortoise" has meaning only when talking about time.
I stopped reading your comment at this point because it is clearly based on a complete misunderstanding of my words.
MouthHole The science site does not publish things condemning dark matter. I remember that
A long time ago, maybe even several years, Michael tried to convince me that a piece of news that was quoted
On the "Hidan" website that with a probability of half the particle of the dark matter was discovered, it is news
scientifically. I don't remember if we intervened or not (maybe Michael remembers), but now it's clear that she knew
It was complete nonsense, by the way aware of the discovery of new particles at the level of a number
Sigmas are not in single percentages. The knowledge that dark matter has been discovered has a confidence level of fifty
A percentage ("either yes or no as I joked at the time) that is obvious to anyone who beeps yes
It was published in the science, but claims that it "still" has not been found are not published... the science
Publishes only news that represents the scientific mainstream and does not seek to doubt any of it
It's in this line of thinking (on the other hand, science does advance by making progress and challenging existing theories).
Michael:
a) There are at least two options:
1) The speed of Achilles is one thousand km/h and the speed of the tortoise is one hundred km/h.
2) The speed of Achilles is one hundred km/h and the speed of the tortoise is ten km/h.
Thus the speed is not defined and it is impossible to know how long it will take for Achilles to reach the tortoise.
b) The series you listed is not related to times. This is a series of distances.
c) In the original paradox, Achilles advanced a hundred meters and the tortoise in the meantime advanced ten meters.
In the example you gave, Achilles advanced ten meters and the tortoise meanwhile advanced one meter.
I made a mistake because I didn't read carefully enough, it seemed to me that it was written that the turtle had moved ten meters in the meantime as in the original.
d) It is not clear what she wanted to demonstrate in the example she gave. Its definition is another series that strives for the same limit. Here, too, the group of points has an upper barrier that does not belong to the group.
Obviously Achilles gets the tortoise. It does not depend on the series of situations. You can define all kinds of series of modes. In all of them there will be a gathering because in all of them Achilles will get the tortoise. But what they all have in common is that the limit does not belong to the field of definition.
To clarify, let's take an example that does not exist in reality:
Suppose that between every two situations there is a third situation where Achilles is on the moon.
This will not affect the considerations and Achilles will get the tortoise.
But in this case it is clear that there is no physical body that can fulfill the condition that between any two states there is a third state where the body is on the moon.
hearing:
If
We treat Achilles as a mathematical point and not as a physical body.
Then
It is not subject to the laws of physics. He can jump to the moon and back in as little time as we want. He can exceed the speed of light. He can develop infinite acceleration and infinite speed.
ב:
The speed of Achilles is known and so is the speed of the tortoise. Otherwise, the problem is not defined and in general it is impossible to claim that Achilles catches the tortoise (maybe the tortoise runs faster than him?
It is customary to present the problem when Achilles runs 10 meters per second and the tortoise runs XNUMX meter per second.
In any case - it is possible to calculate for any given speed and if the speeds are not given then it is not clear to me how you have come to terms with the results quoted over and over again - 10 11 11.1 11.11 11.111 etc.
The sentence you quote from my words does not show my error - partly because this error does not exist.
It is true that the sentence appears in the original paradox. So, by the way - also Achilles. Are you claiming that I'm not allowed to use sentences from the original paradox? This is a paradoxical demand!
In the example I gave, Achilles will indeed catch the tortoise - after all, he runs ten times faster than him - in general - the whole race is that of the original problem, so do you think that the fact that I choose to refer to a certain series of events within the race prevents Achilles from getting the tortoise? I have many abilities, but telekinesis is not one of them.
Miracles:
Just because you say something over and over doesn't make it true. He was and remains wrong. Zenon's problem is related to the fact that the column converges and the period of time in which all the events that Zenon talks about take place is finite.
I don't mean to address your and Thompson's faults but I have a problem with you turning those faults into an argument in a debate.
I also get people who are wrong. I never said otherwise. I just don't get their point. I can, can't I?
I don't want to convince you that the idea of discrete time is wrong. Time may be discrete. All I'm saying is that Zenon's problem does not require time to be discrete.
It could just as well be that Achilles is a caterpillar that can't get the tortoise. Even if this claim is made, I will not claim that it is wrong.
Both of these claims can be true, but they do not derive from the paradox.
Regarding the strange problem (literally) I assume you understand that it is not defined because the first point is not defined.
As soon as you complete the definition of the problem I will be ready to address it.
Michael
If you'll allow me, I'd like to slightly change the paradox of Achilles and the tortoise and hear your opinion.
Let's assume that Achilles should reach the turtle at a distance of 1024 meters from the starting line (of Achilles). The change I propose is a relay race. At the 1024 meter line there will be another "Achilles" and he continues the race.
But, just to add a bit of interest, let's add a few more messengers (not many). Women one in 512 meters, one in 256 meters, one in 128 meters, one in 64 meters, one in 32 meters....
A little patience Michael, as soon as the last one is in place we will start the race! what are you saying?
Michael
I say again and again, Zenon's problem is not related to Tor's limit. The problem I see is the infinity of the column. It's probably a birth defect, but I have a problem understanding that it is possible to end a series of infinite events. Thomson has (had…) the same handicap.
So maybe you will accept the difference and understand that I have a personal problem, which cannot be cured by exercises in Infi'?
Instead - convince me what is wrong with the idea that time is discrete.
Michael:
a) Achilles' speed is unknown. The turtle's speed is unknown. Therefore it is not clear how you calculated the time.
b) The series you are talking about is a series in which Achilles advances every time one tenth of what he advanced the time before it. This is not the series in question in Paradox.
This sentence, which is a quote from your words, shows your error:
"By the time Achilles passes this meter, the tortoise will pass another 10 cm." This sentence belongs to the original paradox.
In the example you gave the distance the second time is 91 meters and not one meter. Achilles will pass a tenth of this distance and the tortoise will pass a hundredth of this distance.
In any case, we know without any connection to the paradox that if Achilles is faster than the tortoise he will catch up with the tortoise at some point.
Apparently she meant it
Give an example in which Achilles takes a series of steps that aims at one limit, while the tortoise takes a series of steps that aims at another limit. When the limit of the Tortoise series is greater than the limit of the Achilles series.
And indeed in such a state of affairs Achilles will not get the tortoise.
But these data are fundamentally different from the data of the paradox.
The paradox speaks of the fact that the series of the tortoise and the series of Achilles strive for the same limit.
Even in the example where the boundaries are different, the time segment is blocked and the block does not belong to the segment. But the question of whether or not the block belongs to the passage has no bearing on the question of whether Achilles will or will not get the tortoise.
incidentally:
Different limits are possible only if the speeds of the tortoise and Achilles tend to zero.
However:
In the paradox data the velocities are constant.
Uncle:
I would like to moderate the conclusions that may arise from my previous comments.
Mathematics deals with the creations of our imagination and in this sense replacing the meaning of time or space in the model is not a solution to the paradox in which time and space are of a different kind.
What is true is that a model dealing with time and space may be inconsistent as a result of assumptions it has about the properties of those dimensions.
A discovery of this type will turn out to be a paradox and it may also be interesting in terms of its implications for physics (just as every mathematical discovery is interesting for its implications for physics).
If Zeno's paradoxes were insoluble within the original assumptions about time and space - this would be a very interesting mathematical discovery.
The point is that they are solvable within the framework of the existing assumptions and precisely in this sense, the escape to another model of time and space in the absence of a paradox (in addition to the fact that it causes us to reject a model that may be exactly the model of reality) may thwart our ability to discover real contradictions in this model.
The long and winding road
http://www.haaretz.co.il/news/science/.premium-1.2154405
Miracles:
From your words to Yossi Simon, it becomes clear that you still do not understand my main claim.
I repeat: the model of discrete time and space is also a mathematical model.
You yourself only deal with mathematical models.
Mathematics allows us to deduce what happens in that model and as I explained - a continuous model - the one according to which Zenon's problem was formulated - does not create a contradiction (that is - the contradiction that Zenon thought he saw is only the result of his misunderstanding of the subject of converging series but it does not exist in this model and whoever understands the Convergence of the time series realizes Zenon's mistake).
I say that attempts to "solve" the problem (which does not even exist in the model of reality that Zenon assumed) by claiming that the model that Zeno assumed is wrong (such as claiming that time is discrete or that Achilles is a caterpillar that cannot reach such speeds at all) are nothing more than evading the problem.
The paradox - as I have said many times - is a matter of logic only. From the two of us - it is you who is constantly trying to connect mathematics with physics and to draw from the fact that Zenon did not know how to calculate columns the conclusion that time and space are separate.
Uncle:
No - and there is more than one reason for this:
1. Mathematics deals with deriving conclusions from assumed axioms. It does not deal with the question of whether the axioms they assume are true. It always accepts the laws of the hakish as true and the axioms of the mathematical structure being analyzed, it accepts when dealing with the same structure. The concept of time, for example, is not part of mathematics at all and therefore its various definitions have no effect on it.
2. As I mentioned in the article - for almost every practical need - it is actually correct to see mathematics as a priori knowledge (as everyone does, with the exception of Abdach's reservation in the article I pointed to) but if you want to examine it as a science, then what you need to do is to test experimentally if its conclusions are correct. Such an experiment can be practical - in front of the physical world (for example, to show in an experiment that if you put a ball in an empty box and then another ball and that's it, you get a box with two balls) or through thought experiments in which the only thing that is tested is mathematics, i.e. - those that make sense The assumptions that are made in them apart from the mathematics are verified truth and then - if contradictions are found it can be concluded that something in our mathematical theory is wrong. This is what happened, for example, with Russell's paradox that allowed us to clearly understand that self-teaching is a wrong thing.
B:
You are wrong in the calculation.
In the section in question, which amounts to 1.111111... Seconds, meaning a second and a ninth, meaning ten seconds less than the time it would take for Achilles to get the tortoise - Achilles will not get the tortoise.
In this series of events, the matter of the closed section cannot contribute anything, and in contrast, the fact that the section is blocked contributes everything.
Michael:
In the example you brought the distance between Achilles and the tortoise at each stage is the initial distance (100) multiplied by nine tenths plus the initial distance (100) multiplied by a hundredth. That is, the distance in each step is a hundred times [(91 hundredths) multiplied by the number of the step]. (The initial stage is stage 0).
That is, the distance between them at each stage is a series of numbers that aspires to a limit of zero.
conclusion:
Achilles will get the tortoise.
That is:
The time segment where the tortoise is in front of Achilles is blocked.
The block point does not belong to the section.
All in all, nothing significant has changed from the original formulation of the paradox.
Suppose that at the blocking point Achilles disappears without leaving a trace.
Nothing has changed in the problem data.
At any time before the blocking point Achilles exists and the tortoise is ahead of Achilles. At the point of time of the barrier Achilles is not found.
Can we say that Achilles catches the tortoise?
point
Did you become a Perminds?? 🙂
ב
It is not correct to say that "in reality Achilles catches the tortoise".
It would be more accurate to say that in our view Achilles catches the tortoise.
In reality there is no movement and change.
Yossi Simon
Zenon does not need to prove that it is not possible to pass through an infinite number of points. Let's be precise - going over an infinite number of points, one by one. It goes without saying - from the moment we agreed that there are indeed infinite points. The problem is not the time. The problem is that something is wrong with the assertion that there are an infinite number of points, or perhaps with the meaning of the concept of infinity. Michael Rothschild gives a certain interpretation to infinity as the value of a limit. This is what they do in Hadoua and there is no argument there. The question is whether it is permissible to throw a mathematical calculation into the physical world - Michael is sure that it is, I am not sure.
But - I proposed another solution, and that is that there are no infinite events. Time and era are discrete and therefore there is a final event. All such "digital" also solves many other problems, except for a large number of paradoxes (Achilles and the tortoise, the arrow, Thomson's lamp and so on). Except for disdain and slander, I did not receive any factual reference to this....
Michael
As you present things from your point of view I certainly understand the apparent inconsistency of using physical assumptions for logical arguments.
But since in your article you yourself strive to raise mathematics from the logical level to the physical one, don't you?!
It seems to me that you believe in mathematics as having a real existence of its own, as a science in itself.
It is not clear to me how this happens and if it is even clear to anyone. (perhaps except from Gmark as I think)
But if we assume that this is true, then there is no problem at all in using physical assumptions for seemingly pure logical arguments.
Because in the situation you seem to be striving for, logic ceases to be pure and detached.
Is not it?
Joseph:
I don't know what exactly you're trying to say, but I'm not that interested anymore.
As far as I'm concerned, I gave the paradox a solution and I also explained to you why what you say is not a solution.
Even endless repetition of things will not prove that another endless repetition will help.
ב:
I thought of another way to convince you that the closure of the passage is not what is at play here.
Let's say that Zeno would have phrased it a little differently and instead of talking about the passage of the first 100 meters (the furrow that Achilles gave to the tortoise) he would have talked about the first ten meters only.
He would say - in order to get the tortoise, Achilles first has to go the first 10 meters.
By the time he passes them, the turtle will have passed a meter (and the distance between them will be 91 meters).
By the time Achilles passes this meter, the tortoise will pass another 10 cm.
Until Achilles passes those ten centimeters….
In short - Zeno will make exactly the same mistake and conclude that Achilles will not be able to get the tortoise and in this case the openness of the passage will have no relevance because even at the limit of the series Achilles will not get the tortoise.
Michael Rothschild!
The basis of Zenon's claim is that there are infinitely many discrete points and this is really true.
For each point I can specify the exact time it passes.
Zenon does not prove the connection that passing through infinity of points precludes passing through infinity of points.
Zenon also cannot claim that it is possible to reach point A while using that point B can be reached.
And so he must prove why infinity of points prevents the first step? And in the end his proof is wrong.
There are many riddles that the human mind and many people fall into, I can name a few of them. (The puzzle of the ant and the elastic rubber tube)
Incorrect, in:
In any case where you try to draw from what happens in the blocked section a conclusion about what happens beyond it - you are wrong.
When they say that Achilles will never get the tortoise, they do exactly that and it doesn't matter what time in that "never" they are talking about
Uncle:
I think you're wrong.
The point of view you present is in my opinion wrong.
And yet I ask you to tell me what in your opinion is the justification for presenting physical assumptions as a solution to a logical problem.
Michael:
The problem is not whether the section is blocked or unblocked.
It is clear to everyone that the section is blocked.
It is clear that on one side of the barrier the tortoise is ahead of Achilles and on the other side of the barrier Achilles is ahead of the tortoise.
The question is what happens at the blocking point.
That is, does the blocking point belong to the segment described in the problem.
The paradox only arises when you try to mistakenly associate the blocking point with the blocked section.
Michael
Precisely after reading, I get the impression that you would not have stopped the discussion you are engaged in there
Especially in the article
"Logic and mathematics versus science - what is common and what separates?"
You would have come to the point of view I raised above yourself.
B:
I think you understood most but not all.
The question of Achilles and the turtle has nothing to do with the question of "closing the section" but only with the question of it being blocked.
Even in a reality where Achilles could jump (in zero time) at ten meters per second, he would catch up with the tortoise (in the sense that after a certain time he would be further away from him), even though he would never meet him.
Uncle:
And to the point: do you reject the definition of the paradox as a logical/mathematical phenomenon?
And if you don't deny her - what justification do you find for "solving" the paradox by making assumptions about the physical world?
There is only one truth.
Truth is absolute.
Science strives to discover the truth.
Paradox is not about truth or falsehood.
Paradox is about misunderstanding.
In reality the truth is that Achilles catches the tortoise.
Anyone can see it.
Each experiment reaffirms this.
The paradox of Achilles and the tortoise stems from an error in thinking.
The error in thinking is treating an open section as if it were a closed section.
In the open section there is a match between points where Achilles is and the natural numbers.
In the closed section there is no adjustment.
The match is not defined at the endpoint of the closed segment.
Uncle:
I assume you wrote the previous comment before reading the articles I directed you to
Michael
I'm sure you'll continue, that's what's great about the approach I mentioned.
Everyone not just you can continue to breathe or sleep.
I also agree that there is no absolute truth in the claim that there is no absolute truth exactly as you stated.
It definitely leaves the claim in place come to think of it.
The pursuit of self-truth is unshakable. replaced by the developmental benefit of thought in an approach of multiple truths and multiple solutions.
And this is not anachronistic philosophical nonsense. but in the potential benefit of the approach.
Uncle:
I suggest you read two articles I wrote on the subject of your last comment:
1. Is science just another religion?
2.Logic and mathematics versus science - what is common and what separates?
Uncle:
Here I reject your approach completely.
I inform you that I will continue to breathe even if they show me that this is an outdated way of life that is already hundreds of millions of years old.
Is the claim that there is no absolute truth an absolute truth?
In short, Yossi Simon, you "solve" the paradox by refusing to address the arguments put forward by Zenon.
great "solution"
Michael
In my opinion the old mathematical/philosophical approach that ascribes absolute self-truth to logical objects.
And to a significant extent this also applies to the judicial interpretation of reality.
It got old and lost its cool.
In my opinion the attitude of no single absolute truth has already replaced the old attitude even without it being explicitly announced.
Modern logical truth is a puzzle of elements that fit into a certain picture according to need.
These elements should not be attributed any uniqueness or sanctity of absolute truth. Some or all of them can be replaced by others.
And yet the final puzzle represented the same picture.
The old attitude that attributes absolute truth to things is a serious obstacle to the development of thought.
The origin of this approach is in the various religions that force a one-dimensional and limiting perception in order to control the mind
the believers Supporting the absolute truth approach is a limiting approach. and is not worthy of modern mathematics.
Because in the end it also turns mathematics into a religion.
And therefore I disagree with you on the point of the importance of the "correct" solution to the paradox.
The role of paradox is to open the door to multiple alternative solutions and multiple alternative truths.
And not to stay in the anachronistic approach of absolute truths.
Take Darwinian evolution for example.
Would you say that nature tried all kinds of experiments just to find the one truth that would lead to the creation of life.
Of course not! Life could have been created in many forms and this can be proven from the great variety of life that exists.
It is therefore clear that we must be at least as wise as nature and not attribute absolute truths to the logic we invent.
To Israel Shapira, Michael Rothschild!
Now we will move on to the failure of the paradox
1. A paradox is defined as - ((A drags B ) and (A drags B not )) .
2. In Achilles there is a drag only to one side "seemingly" of (B not ) movement = B, therefore it is not a paradox by definition.
The failure itself
3. Movement is related to the world of concepts of distance, time, speed, acceleration, direction
4. The distance of a single point is zero, and likewise the sum of an infinite number of discrete points
5. Therefore, the number of discrete points that pass is not important because the transit time is zero.
The whole issue of internal series is irrelevant
It's all
1) The time point at which Achilles obtains the tortoise constitutes an upper barrier to the time segment mentioned in the paradox.
The upper barrier is not included in the group of times described in the paradox.
The segment of time described in the paradox is an open segment.
Therefore, within this time segment, Achilles does not get the tortoise.
2) The match that describes the distance of Achilles from the turtle is an indefinite match in the closed section, it is defined only in the open section.
3) We know that Achilles catches the tortoise only because we know that Achilles' motion is continuous.
We could for example define the same problem with the same data but with a discontinuous movement of Achilles.
If, for example, we were to decide that at the upper limit point of the time described in the problem Achilles is one meter behind the tortoise.
So it wouldn't have changed any of the problem data. The entire description would have remained the same except for what happens at the upper blocking point of the time described in the problem.
Michael
You write: "Israel:
I don't know what the disagreement is between us."
Re-read my comment, the dispute paragraph was directed at miracles.
Miracles.
You write: "1) Zeno was not talking about the sum of an infinite series, he was talking about the infinity of an infinite series. This is my opinion."
OK, Rachel the daughter:
Do you accept that according to Zeno and Paradox …..10+1+0.1+0.01+0.001 to infinity or at least to a large number like 1000 is equal to infinity or certainly more than 12?
And do you accept that he was wrong, meaning that no matter how many members we add to this expression we will never reach 12?
Michael Rothschild!
Beauty! I am happy that finally someone understands me and justifies me.
Now let's continue! And summed up the paradox.
The argument is that it is not possible to go from point 0 to a point greater than zero X, so that point X is smaller than any point greater than 0 (topological issue). Where is the justification for this claim? (the missing part about induction)
Yossi Simon
Before my friend Michael jumps on me again, I must point out that using induction is problematic. The first reason is that mathematical induction is an axiom. And we already learned a long time ago that blind faith in axioms is problematic.
The second reason is called Goodman's paradox. Read about it……
Israel
1) Zeno was not talking about the sum of an infinite series, he was talking about the infinity of an infinite series. This is my opinion
2) I don't think he's wrong. But again - he did not talk about the amount of the series. This is my opinion.
3) No - the paradox is not resolved in my opinion. That is, knowing how to calculate the limit of a convergent engineering column does not (in my opinion) solve the paradox.
4) Here you say something interesting. We don't know much about Zeno, and what we do know is Plato's letters - Plato was born after Zeno died. We do know that Zeno was a student of Parmenides, and Parmenides spoke (again, through Plato) of a concept called "monism".
I think the paradox is not resolved. And I offered a possible solution (of course - the idea itself is not mine....)
Yossi Simon:
As mentioned - I already answered all of this in my words to David.
You are not wrong (or rather - my dear Shoshani is not wrong) in your conclusions.
He is only wrong in that, instead of showing the source of the error, he only shows that Zeno's (wrong) considerations lead to further paradoxes.
Michael Rothschild! Another thing
Let me present the paradox as follows:
Two cars are moving on a straight road. During the journey, the maximum speed of the first car is always less than the low speed of the car traveling a certain distance behind it.
According to the arguments of the paradox the second car will never overtake the first.
So which gathering/entertainment column are you referring to?
Likewise, regarding the time between hour 1 and hour 2, the time should pass in 1 and ten minutes 1 and a hundred minutes and another root of 2 seconds, etc., so to speak, we will never reach hour 2.
And the argument that it is impossible to skip an end of points each of length 0 is not a valid argument
Michael Rothschild!
Prof. Yakir Shoshani refers to this problem, I cannot give a reference because it is not on the Internet,
to the root of the problem.
Let's start from the beginning. Follow step by step and please point out where I am wrong.
1. Suppose that after 180 meters Achilles catches the tortoise.
2. So the paradox tries to claim that Achilles is not able to go 180 meters.
3. In this case, the carrying of the tortoise is unnecessary and the proof is reduced to the fact that Achilles is unable to pass 180 meters
4. If Achilles is not able to go 180 meters, then he is not able to go 70 meters, or one meter or a millimeter.
5. In other words, Achilles cannot move at all because at each step we can find numbers that prove that Achilles is unable to reach this point (due to the fact that the length of a point is zero and any distance greater than 0 contains an infinite number of points) meaning there is no possibility of movement, continuity, etc.
So far it's Prof. Yakir Shoshani who clarifies the dilemma they were dealing with in the past.
The proof is based on a similarity of a proof by induction, but does not include all its components, the argument that a concentration of infinite points prevents movement is not a valid argument.
Israel:
I don't know what the disagreement is between us.
1. I don't know what "according to Zeno" is, I don't know how to read Zeno's thoughts, but apparently he didn't know the concept of the column and didn't think anything about it. This should be said to his credit, especially in light of the fact that nowadays there are people who do know the concept of the column and still make mistakes like him.
2. As mentioned - there is no answer to the question because Zenon did not think of a column. He assumed that if there is an infinite number of events then it must take an infinite amount of time. He was wrong about this, as mentioned, because he did not know the concept of a column and in particular he did not know about converging columns.
3. The original paradox has been resolved and I voted in this discussion on my article in which I described the solution.
4. We are discussing the same paradox. We simply know how to analyze the situation better than a buffet.
It is true that in my words I separated the column and the sum of the column, but it is really immaterial.
What is important is the way in which what I called "the sum of the column" is calculated and what is here called "the column" and the legitimacy of this method of calculation also to represent nature
Miracles:
Probably nothing really will help.
How do you know Achilles will get the tortoise?
Have you seen Achilles?
did you see the turtle
Are you a fortune teller?
Let me help you with the answers: negative urine.
You know he will get it because you are running the predictions of a mathematical model of nature.
You understand?
Probably the answer here too is negative.
Quote from wikipedia over and over again my words.
Indeed - the column is a sum of series members - just like I said. Why do you find it appropriate to repeat this and what is the logic behind such use precisely in order to show that I was wrong? odd.
But it is.
I realized that nothing came of this argument.
Michael
You are right about the definition. I unknowingly sinned in demagoguery.
Essence: As mentioned, if my memory serves me correctly, the answer 1 or 0 appeared in a book where it was also said that Leibniz was wrong when he said half.
The book was written by a contemporary mathematician.
If we go by the assumption that I remembered correctly (not sure), then there is at least one contemporary mathematician who claims that the sum of the column is 1 or 0. If you really insist, I will do a more thorough house check and maybe I will find the break.
Miracles
Your reservations are white in Avron's book. But why go far? It is enough if we look at the "law of the second law":
"What kind of completeness can one expect from a Torah whose cornerstone is the forbidden division by 0? Release the law immediately, along with a letter of apology for the prosecution's ignorance."
To try to resolve the dispute between us, simply answer the following simple questions:
1. According to Zeno, can the sum of an infinite column converge to a finite sum? Yes or No.
2. And if the answer is no, was he wrong? Yes or No.
3. And if he was wrong, is the original paradox resolved? Yes or No
4. And if the original paradox is resolved, aren't we now discussing a different paradox that may be only indirectly related to the original one? Yes or No.
Working.
Michael
Always what you do is right. God bless you.
The paradox is a paradox precisely because, after all, Achilles wins. It's really not a paradox in logic (like Thomson's lamp example).
I am quoting again from Wikipedia - "in mathematics, the concept of the column comes to denote the sum of a series". I see that you have a hard time with this - so I will bring another quote "In mathematics, a series is an ordered list of objects, called the members of the series"
You are the one who likes quotes……
Maybe you will relate to the content of what I say instead of stinging?
Miracles:
You accuse others of arrogance and when the same blame is (rightfully) directed at you, you define it as a personal downfall (once two people who were fighting were caught by a policeman, so one of them told the policeman "it all started with him giving me back"). interesting!
Even the unfounded accusation that I am getting down to personal lines is getting down to personal lines.
I keep giving factual reasons and you keep ignoring them.
Including your return in this response to the fact that I failed to explain in one word why discrete time is a wrong idea, although I repeat that discrete time is an idea that must be tested physically, but it simply has nothing to do with the paradox that is a phenomenon in the field of logic.
You are quoting passages from Wikipedia that match exactly what I am saying and trying to use them as an argument against what I am saying.
A column is not a series of partial sums.
A column is an ordered collection of numbers (which, among other things, can be referred to as its partial sums which constitute another series).
The limit of this series is the only value that can be considered as the sum of the column and this matter, beyond its internal logic it also works in drawing conclusions about the world and much of physics is based on it.
This is a bit of a joke about the Jews and their letters, once upon a time and maybe I can reproduce I knew the proof of this
Water blowing
who said that?
It is possible to go below "A0" and that is the mistake
Water blowing
You are the joke here. In mathematics there is a world full of infinities. There is 'A0' and there is 'A1' and there are also 'B' numbers.
There is no joke here.
Where's the mistake?
There is a common mathematical error that calls the first order of magnitude of infinity "a". This is a bit of a joke about the Hebrew language and the Jews, there are orders of magnitude infinitely small and as small as we want
There is no small and small end and no big and big end
Water blowing
Reverse Gota, Reverse.
On a large scale the world seems continuous to us. Precisely on a small scale the world seems discrete.
Why did you write this?
I will try to illustrate how a thing can become smaller and smaller and strive in the physical world.
If parts also move in other dimensions, they are in the dimension you count them aiming for the statistical length of the place in the limited dimension in which, again, you count them.
And if their movement in the other dimensions can build and deplete them in the opposite direction, then their value can strive and the emphasis on striving and their actual size in the limited world can go infinitely smaller and smaller. Thanks
Dear Nissim
The physical world may be discrete but in small and small parts with no end and therefore also analog
Another different point of view from the direction of the theory of relativity.
According to the theory of relativity, all things always move at a constant speed (the speed of light) towards the future. Changing the direction of the speed (for example giving an additional spatial speed) will reduce the vector towards the future and therefore time will appear to slow down.
Zeno talks about a static world. Our world is not like that. In our world, things always move at the speed of light.
point
An interesting approach. I'm trying to see if your approach is similar to mine. On the face of it, I don't think so. What you're saying is that you can't actually define a position precisely, so we don't have infinite steps.
So maybe it's similar 🙂 In the end you reach plank-sized steps and these steps are discrete.
In any case, I'm glad that someone else is brave enough to understand that there is indeed a significant paradox here.
Touch
The mistake (ie the wrong assumption) in Zeno's paradoxes is that Zeno assumes an actual, realistic world, in which an object is well represented. But what to do and our world is not like that.
Our world is a world that responds to quantum mechanics, and as such, the object does not have a well-defined place but is spread, and then all the paradoxes fall away because the spread itself exists in several places in space at the same time and then the movement is nothing but the change of the weight of the spreads across the space.
Those smears, it is not possible to talk about them in the actual Achi language, they are defined by imaginary numbers.
In short, Zenon is right that in this world that he created, movement is not possible, but this is not our world.
Michael - who talked about you? 🙂
I will quote from the source you love so much, Wikipedia, one sentence "If the sum seems to be "getting closer" to some finite number, it means that the column is converging."
If you want to be precise - a column is a series of partial sums of members of a series. The amount strives for the limit but never reaches it. I don't see how you can argue that. You can define the value of the sum as the limit of the column, but you cannot draw a conclusion about the world based on a definition.
I didn't say that everyone is wrong and I didn't say that I was right. I tried to develop, for those who are interested in hearing, a less familiar line of thought. I'm not the only one who thinks differently than "everyone else". I thought about these paradoxes, and I really think the solution you keep coming back to is beside the point. You did not respond to the content of my words, you simply laughed at my lack of understanding. You always manage to get down to personal lines and the whole discussion here loses its value.
You failed to explain in one word why discrete time is a wrong idea (regardless of our paradoxes). You just gave a silly analogy that has nothing to do with what I said.
Michael - it could really be that the idea is completely wrong. I would love to hear why and learn. I was able to find a number of supporters of the idea (let's be precise - it's their idea and not mine), such as Konrad Zoss, Stephen Wolfram and Edward Fredkin.
And just an anecdotal detail - "Zenon machines" are a familiar concept in computer science and provide a certain model of hypercomputation (which is part of my research field).
Miracles:
The arrogance is yours when you point out that everyone else is wrong but you.
The confusions are also yours and this is expressed throughout, including in the last comment when you say that the value of the column converges.
Value is value and value does nothing but be itself. In particular, it does not converge. As explained to you - the series of the partial sums of the column converges.
As you know, it converges to a number and nothing else.
This number is defined as the sum of the members of the column and it is a logical and consistent definition to the extent that there is no other number in the world that can be defined as the sum of the members of the column.
I already mentioned Thompson's paradox in the explanation I gave to Israel about a question they ask about a point in time for which no data was given. You can call it Super Task or "O-Brain" or any other fancy name and it won't make what happens in point 2 definite. There is no way to derive the answer to the state of the lamp at a certain point in time from a description that only refers to other points.
This is exactly the mistake I pointed out in Zeno's attempt to deduce what happens at the points that his series of events does not reach (not because it is impossible to reach but because he chose to describe this series and not another).
The solution "time is discrete" is likened to the solution "Achilles is generally a snail that cannot catch a turtle".
It's just not relevant but I've already explained it and our hope is lost.
I think that Zeno's paradoxes are not as resolved as is commonly thought. It is common to think that they are solved - because we have a calculation that talks about the concept of the limit. I'm talking about Hadova, of course. This calculation causes us (some of us) to get confused between the concept of the limit and the concept of a function value at a point. One example is Zeno's Paradox of the Arrow. Here we are confusing the value of the function x/x when x->0 and 0/0. A second example appears in the paradox of Achilles and the tortoise: here we confuse a series with a series. I will explain: the converging column value 1+1/2+1/4+.. converges to 2, and we define this value as the sum of the series. I don't care about that (although I don't think it's right to confuse 2 different concepts, but never mind). What I care about is expressed in another paradox - Thomson's lamp.
Thomson gave a good example of a super task. The example is a lamp that you turn on, wipe after a minute, turn on after half a minute, turn off again after 15 seconds and so on. What is the state of the lamp after 2 minutes?
Obviously, the answer is not "on" or "off" (reminds me of Grundy's column). The question is what is wrong with our way of thinking.
Despite the arrogance of some people, I think there is a lot of room for thought here. And this is also true of Zeno's paradoxes. One of the problems raised by the paradoxes is a problem of measurement - how exactly can you check that 2 minutes have passed? Is it possible to measure time with infinite precision?
I think there is a solution to this - I think time is discrete. It solves the three paradoxes and I don't see it creating new problems.
Is there anything wrong with this idea?
Israel
I wrote you a comment yesterday but it disappeared. At the risk of repeating things, I will explain my understanding again.
Israel:
Regarding the amount of the column being defined or not - there is simply a development of the language here.
In contrast to Euler's time, today, the word "defined" regarding series sums is already defined.
that's it.
Therefore, there will be no change in the future and the question of whether the amount of the series is defined or not will remain one answer. It will no longer be a matter of "Euler's opinion" or "Bernoulli's opinion".
After all, if you delve into the question, it will be clear to you that in the current definition of the sum of a series - a definition based on the definition of convergence, even Euler would have said that the sum in the series you described is undefined.
Uncle:
First of all, I like to note that I get the impression that you, unlike some other commenters who just make statements or bring other people's arguments without being able to stand behind them, understand what we are talking about (from reading other comments I got the impression that ב already understood the idea, even though he mixes the nomenclature between "the data of the problem" and "the data on which Zenon's erroneous judgment is based" and of course thatIsrael understood already quite a long time ago).
In fact the dispute you describe is a dispute of emphases and this is where we really disagree.
To me, dealing with paradoxes in the logical sense of the word is the most important thing because, unlike just solving a problem that brings us from a state of "not knowing the answer" to a state of "knowing the answer", solving the paradox brings us from a state of "knowing something is wrong" to a state of "Knows a true thing".
The specific dealing with the term time (which is a natural phenomenon and not part of logic) is of course important, but there is not much point in it if people do not know how to consider logical considerations.
Specifically, in the question we are discussing, Zenon's argument is not at all based on the continuity of time or space (he is merely talking about a series of discrete events).
What it is based on is the mistake of drawing a conclusion that refers to the entire timeline from a collection of events that by definition are reduced to only a finite part of it.
The issue of continuity (in itself, without any relation to time) is discussed in depth within mathematics and has clear and effective definitions. There are even types and shades of continuity that mathematics deals with (such as continuity of equal measure, continuity of derivatives of different orders, etc.).
In this sense, time is no exception and the same reduction of the problem that you proposed to the timeline can also be copied to the roadmap and claim that it is impossible to go from point to point.
The thing is, as I said, it doesn't deal with the error. It just proves that it exists. In fact, what these reductions (if we are not discussing the convergence of the columns) say something like: "If it is impossible to reach 11 o'clock and nine seconds then it is impossible to reach any hour" or "If it is impossible to reach the meeting point then it is impossible to reach any point" but they do not indicate For the mistake that led to these wrong conclusions.
It is similar to a person who comes to his friend, shows him something and asks "What is this?"
"I don't know what it is," the friend replies, "but here's another one"
Who underestimates? Did you hear a screeching sound?
I just don't understand what the problem is. It was made clear again and again that the mathematical problem was Zeno's ignorance of the convergence of the infinite series to the end. This was supposed to be the end. So maybe you can explain why you don't finally accept it?
Everything else, quantum steps, etc., is not related.
Israel
I don't understand what you don't understand. The sum of the column is not 2. This is the definition of an infinite column. But, that's not the problem here - the problem is that there are endless events. And this means that it is not possible to reach the end of the event column.
Israel - stop belittling, and bring a logical counterargument. I offered a solution to the problem, you didn't respond...
ב
If you believe in the big bang, that is, the entire universe with its galaxies, nebulae and zebras started from a primordial atom of size zero - believe in everything.
Miracles
I understand your (philosophical) argument like this: a distinction must be made between 2 mathematical, i.e. the sum of the column....1+1/2+1/4+1/8 and between 2 physical where the two are better than the one.
And for that I only say:
If a young heart ever suffered
Dangerous false doctrine
Same smell and taste
Old witnesses will not evaporate.
Israel
You and I understand Zeno differently. In my opinion, he was not talking about an infinite column. Our infinite column is not equal to 2, it strives for 2. In my opinion, the problem is that there are infinite events, and the meaning of a series of infinite events is that there is no final event, meaning that the series will never stop. The fact that the distance column tends to 2 is really fascinating, an amazing mathematical exercise, but does not solve the paradox.
I say - if our model creates a contradiction, maybe it's worth changing the model. I claim that the world is discrete. There is a quanta of every measurable variable, such as time and distance. This completely resolves both of Zeno's paradoxes (the other being the arrow paradox).
The claim may be strange, but that is no reason to dismiss it.
The essence of Achilles and the tortoise.
I guess Zenon had other things going for him.
not summarized:
The use of mathematical models to describe physical reality.
And in particular:
The Big Bang theme.
sounds deep.
But it still seems to me that Zeno's claim was that the sum of an infinite column is infinite.
And he was simply wrong, and this was due to a lack of sufficient knowledge of the nature of Tories.
Zenon essence.
Israel:
In the framework described in the problem, Achilles will not reach the tortoise in a finite number of steps.
hearing:
In the data frame of the problem, Achilles will not catch the tortoise.
The point at which Achilles obtains the tortoise is outside the scope of the discussion of the problem.
This point is well defined.
Check the definition of cut (Dedekind cut).
The mathematical description of the problem of Achilles and the tortoise defines a point where the description is not valid. This is the point where Achilles catches the tortoise.
It is just like a cut in the rationals can define the "square root" but the "square root" does not belong to the cut because it does not belong to the rationals.
All we can know is that at a certain point in time Achilles does catch the tortoise and this is because if there is any distance between Achilles and the tortoise then it cannot be at this point in time but only at some point in time before that.
Again I emphasize:
The time point at which Achilles catches the tortoise is outside the range described in the problem data. It does not belong to the set of points described in the problem.
Michael
The adherence to a precise form of analysis as you require is apparently necessary in the discussion of this type of issues.
I admit that I did not bother enough with my argument.
In fact, my intention is mainly focused on the thought process that underlies the logical sequence of the presentation of the paradox.
In my opinion the solution of the paradox itself is not that important. Like the importance of deciphering the meanings of the fundamental elements "time" and "continuity" which stand up to careful examination through Zeno's paradox.
The solution you gave to the paradox requires the use of the element of time. and defining a time point attributed to the logical sequence of the description.
But this leaves the element of time itself and its attribution possibilities, in a state of implicit assumptions that are outside the focus of the logical sequence.
Deciphering the element of time and sequence requires, in my opinion, to highlight a number of important features of the implicit assumptions.
1. The appearance of loops of different types and places is a fundamental feature when mixing time and continuity.
2. The loops can be copied from a "problematic" place in the logical sequence to another place that makes sense. at the same oppurtunity
to replace one type of loop with another. After all, the solution of the border and the convergence of columns is a loop in a different dress.
3. The attribute of logical continuity must be included in the attribute of the element of continuity that is placed in an enclave within the logic.
It is not possible to ignore the dependence of logic as a sequence that describes processes on the one hand from the very concept of continuity that is laid down
In the basis of the description itself.
4. Which brings us again more firmly to the above loop feature that becomes a loop within a loop.
And at this point the logic becomes a paradox.
5. And I will conclude with the last feature that must stand out, although not the last one that exists on the subject.
It is the reality-bound existence of meeting points between the attributed elements.
That is, endpoints of different types and in different places with different features.
When, as in 2 above, they can be copied using different techniques from a problematic place to other places.
All of the above, in my opinion, is a necessary introduction to deciphering these elements of time and continuity and their inclusion within
logical sequences.
So when you wrote "he won't reach the turtle in a finite number of steps" you meant steps. break up
But according to the paradox he will not get the turtle at all. And as already noted, the reason is that Zeno's assumption is that an infinite number of steps is physically expressed in an infinite distance that Achilles has to travel, because Zeno's wrong assumption was that an infinite number of intervals of distance add up to an infinite distance, even if those intervals are small and go by a ratio of 1/10.
The discovery of infinite series summing to a finite size solved the problem 300 years ago. So why are we still debating? Mila Nissim who argues with philosophical reasoning that mathematical analysis is not necessarily applicable to the field of physics. Zeno's claim was mathematical, namely that every infinite column sums to infinity, and he would indeed be right in this particular paradox if this particular infinite column sums to infinity, like many other columns. But he didn't.
Israel:
There is something confusing about it.
There are steps of Achilles and there are phases of the problem which are also called steps.
If we talk about Achilles' steps, this is a completely different problem.
In the problem as it is presented, the steps of Achilles are ignored. Achilles can also float in the air without taking a single step.
The problem is about stages in Achilles' progress and not about Achilles' steps.
ב
In the link you brought, it doesn't say that Achilles' steps are gradually smaller.
So if we start from the assumption that each step is a meter long, then in the 112th step he will reach the turtle.
And for each certain step length, there will be a finite number of steps in which he will reach the turtle, except for a Yemeni step which, as you know, consists of one step forward and two steps back.
According to the link, Euler also claims that the amount is probably half.
Be that as it may, the answer is not simple. I remembered one or zero and Sleibenitz was wrong. After I checked it became clear how complex the question is, and even Euler could not come to an unequivocal conclusion. In the book "Zero" by Charles Zeif it is said that Reverend Grundy used the series to prove the existence of God, while according to Paul Hoffman (Ardosh) Niels Henrik Abel called the column "The Devil's Column" and went crazy because of it.
So statistically, even if mathematicians today claim that the sum of the column is undefined, it will probably change in the future.
It seems pretty clear to me that the sum of the column is one, zero, or something in between.
Israel:
http://he.wikipedia.org/wiki/%D7%94%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1%D7%99%D7%9D_%D7%A9%D7%9C_%D7%96%D7%A0%D7%95%D7%9F
Time is up.
The distance is finite.
The number of steps is infinite.
Israel:
I still maintain that it is not defined and I don't think you will find any mathematician today who claims that the sum is defined.
The partial sums are defined but the sum of all the terms is not defined - not as the word "definition" is used in mathematics.
You can define the "group of partial sums obtained when the members are summed according to the given order" and it will be defined (not as a sum, which is a number, but as a group of numbers) but there is no justification for claiming that the sum itself is defined.
ב
According to the original paradox, i.e. the one described in Wikipedia, Achilles will reach the tortoise in a finite number of steps.
Michael
Sorry for digressing. Now, according to the link I provided, Leibniz claims that the sum of the column is half. Bernoulli two thirds. My missing book is one or zero.
Are you still claiming that the column amount is undefined? Style division by zero, undefined. Or is there no agreement on the sum between the different mathematicians?
Israel:
A sum of a column does not exist if it depends on the form of the schema.
Miracles:
What you want or don't want doesn't change anything in the paradox.
Even the fact that you think it is possible or impossible to describe the world using mathematics does not change the fact that all the alternative descriptions you brought (and all the descriptions that the science of physics deals with) are mathematical.
You want to argue that the world is not continuous? Charge! No one is stopping you! Just leave the Achilles paradox and put it to rest because it doesn't testify to it! As I said, the possibility that the world is discrete exists and is rightly investigated, but no sane scientist says "Here! We have the story of Achilles and the tortoise and therefore the question of the world's discretion has been decided!" Sensible scientists don't do this because it's so wrong - just as the parable you didn't understand demonstrates.
Israel:
You choose from the text a single sentence that does not exhaust the meaning of the term and does not even appear first in its description - just to justify the incorrect use of language.
The translation of the term into English is definition. Do you have a confusing pun there too?
The word "definition" is actually related to a fence and in the right contexts I also make use of this fact, but to deduce from this that in any case it is a fence that has room for maneuver is demagoguery.
Uncle:
Your conclusion regarding the stopping of time according to the same considerations as in the paradox is correct.
I thought about it too, but I avoided presenting it for two reasons:
One is that it does not solve the paradox but only shows that it exists in a more basic layer - that of time itself and in this sense, this reference is evading the resolution of the paradox (in a similar way to the one where people who try to avoid explaining the origin of life turn to panspermia).
The second is that from the nature of the arguments that came up in the discussion, I understood that my interlocutors would not understand the connection.
The second consideration is not relevant to our case and here you fall into exactly the repeated mistake of miracles.
The paradox is in the realm of logic and mathematics. Zeno assumed continuous time and space and argued that there is a paradox in this assumption. He was wrong in his assumption, as I have shown that even in continuous time and space a paradox does not arise here.
The discussion about the nature of the world and alternative models for its behavior does not belong to the paradox. On the contrary: it obscures the fact that there is no connection between paradoxes (which are a term from the field of logic) and the question "among the consistent logical models of the world - which is the correct model?"
In the paradox of Achilles and the tortoise:
If indeed Achilles advances as described in the problem
Then
He will not reach the turtle in a finite number of steps.
but :
Achilles catches the tortoise.
conclusion:
There is a limit to the infinite series of states described in the problem.
And we are left with the question:
Suppose that a certain infinite series has a limit.
Does there exist one of the members of the series equal to the limit or does such a member not exist.
And the answer:
Although the limit exists, it is still possible that none of the members of the series is equal to the limit.
Let's go back to Achilles:
If we refer to a point in time when Achilles has already passed the tortoise. (as close to the turtle as we like) . It seems impossible to describe Achilles' progress in the same way as it was described before he reached the Tortoise.
hearing:
The point where Achilles catches the tortoise is a cut on the number line. To the left of the section, its progress can be described as described in the problem.
At the intersection point and to the right of it, Achilles' progress differs from that described in the problem.
please pay attention:
Achilles' progress is continuous. Only the description changes continuously.
conclusion:
There is no flaw in Achilles' progress
on condition:
that we limit the description of progress to the point of intersection and no more.
And since the description fits up to the cut point, it cannot describe what happens at the cut point.
Well, B, so what is the amount of my column? Exists or does not exist? If not, then why does Leibniz claim that it is and that it is half? Is Leibniz also trying to invent new mathematics? and Euler and Bernoulli, and..
Oh, I forgot you don't browse confusing links.
And if it does not exist, then can we say that, as in the case of division by zero, the result can be any number, including which ones are less than zero and greater than one?
Israel:
Are you trying to invent new math?
As far as I know:
If the column has a sum
Then
The amount of the column exists. The members of the column do not arrive. They arrive one by one in order to join the sum. They just exist.
The sum of a column is simply an existing number.
Just like the sum (1 plus 3) existed even before someone came and performed the addition operation.
The same thing happens with the limit of a series.
If the series converges, then its limit exists and is simply a number. This number does not wait for the members of the series to run towards him and gather by him. They have been gathered there since time immemorial.
Michael Israel
I am ready to accept the definition "the value of a convergent infinite series is the value of the limit". But I can't accept that it depicts reality. In the paradox there is a sum of infinite terms, and I don't think that it is possible to connect infinite terms - even if the limit can be calculated!!! That's why I suggested you read about Super tasks.
Second point - I don't think the world can be explained by mathematics. Of course it can be described and predictions can be made with it (sometimes yes and sometimes not).
Third point - I want to argue that the world is not continuous. I don't see any problem with assuming the world is discrete, and I think it answers a lot of problems in a simpler way. Here - I suggest you read about Edward Fredkin.
From "Lesson in Hebrew":
"A successful definition is one that defines a concept"
So the sum of the column I brought is definitely defined. It can be 1/2 as Leibniz claims, but it cannot be less than 0 or greater than 1. What are the definitions?
In the book (I will still find it!) it was written: 1 or 0.
Miracles
There are many ways of handling infinity in mathematics, and even distinguishing between different sizes of infinities.
Axiom of parallels: "Two parallel lines will not meet or meet at infinity".
Even in physics before the big bang theory, the universe and time were considered infinite.
Zenon's "paradox" arose from the incorrect assumption that if a series contains infinite terms, its sum is also infinite, as indeed happens to the sum of many infinite series.
In Infinite series (I don't say "infinite series" so as not to upset B) that were not available at the time of Zenon (sounds like Zen language), the convergence of the series to a finite sum is proved.
Therefore the whole "paradox" stems from a lack of information.
Michael
In my opinion, a logical sequence like Zeno's seems paradoxical because it swallows a number of assumptions that give it validity.
For example, the description of Zenon's movement in front of the turtle describes a relationship between the movement of two objects but ignores the movement of time itself.
After all, it is impossible to talk about movement and ignore the concept of time. But that's what the paradox does, it raises time.
As soon as you notice this, then seemingly the infinite division can be applied to time itself.
and conclude from this that if Zenon stops, time also stops. But then the movement has no meaning at all. Because at what time will time decide to stop precisely at the point where the paradox occurs or perhaps precisely at the first step.
But if time has stopped and the movement does not exist then the whole paradox does not exist either because it is impossible to start the logical sequence at all.
And maybe time changes its speed at will then what happens to the movement.
Added to this is another element which is also placed in the enclave in the description of the logical sequence.
And it is the assumption that relative movements are continuous.
The assumption of continuity is not required by reality at all, perhaps things progress or change not sequentially but in transitions between one collection of situations and another.
Hadoah itself is based on this assumption of continuities, but why it is not possible to get the same results and perhaps even better tools, through the logic of jumping between different situations.
In conclusion, when you check the assumptions implicit in various paradoxes, you get a paradox of paradoxes.
which renders the given logical sequence meaningless.
Because if the answers to the questions are both positive and negative, we will not find the hands and habits in all this.
Yossi Simon
You are the one who does not understand, in my opinion, the paradox of Achilles and the tortoise. The question is not how Achilles wins, we all know he won. The question is where is the error in reaching this conclusion.
To say that a column converges is an unacceptable solution to me. Whoever says forgot what the definition of a converging column is...
It is possible to calculate the trajectory of billiard balls using the Pythagorean theorem - but this is not the explanation for the movement of the balls. (And by the way, it is impossible to calculate the trajectory of pool balls…..unless the world is discrete, see below).
I'm not sure that everything that happens in the world can be expressed mathematically. First thing - I don't understand why this has to be true. Secondly, we know simple physical phenomena that do not have an analytical solution (and it is proven that they do not have such a solution).
But - it has nothing to do with paradoxes. I think there is a solution, and it's not a cheap (and wrong) mathematical trick. I think the solution is that the world is discrete - there is a quantum of time and there is a quantum of distance.
The parable is wrong.
Miracles:
It's not that you don't think too much - it's that you don't think enough.
You didn't understand the parable.
Too bad.
I'm tired
Michael Rothschild
You assume what you are trying to prove 🙂 You assume a continuous world - and assume that in this world movement is possible. Not nice Michael...
I claim (without thinking too much I admit) that time and space are discrete, and that this is the solution to Zeno's paradoxes. I also claim that it is (called it) crooked to explain a physical phenomenon with the help of a mathematical trick (and of course this trick is wrong - the value of a function at a point and the limit of a function are very different things)
Yossi Simon:
The paradox data is absolutely overwhelming to me.
is also Wikipedia They are rivers.
Michael Rothschild!
I believe that the data of the paradox does not flow to you!
I will present the "riddle"
Fact number 1: Achilles ran 3 times faster than Dalton.
Fact number 2: Dalton stands 100 meters in front of Achilles
Data number 3: A chase begins (let's say that at time 0) KA runs at the top of his speed
It is clear to everyone that Achilles will catch up with Dalton after 150 meters. (If a step is half a meter long, for that matter, then only 300 steps are required) (Achilles does not know the riddle at all and has no reason to shorten his steps)
In the "riddle" they try to present a logic that leads to the conclusion that Achilles will never get the tortoise (Dalton) according to the following arguments:
When Achilles goes 100 meters then the tortoise has moved a little, 33.3333 meters. When Achilles reaches this point, then the tortoise moves forward a little more, and God forbid returns. And the claim that every time Achilles reaches the new destination the tortoise moves forward a little. And the question, what is wrong with the logic?
And if the arguments are correct, it follows that Achilles is not able to go 151 meters because he is not able to go 150 meters.
But 150 meters are the puzzle data. For each selected distance we can find data leading to the fact that Achilles will not cross the above distance. That is, there is no movement at all.
And according to the spirit of the "riddle" it is possible to define similar puzzles in solving equations in mathematics and in any subject related to sequence.
And the subject of borders and series is not relevant because there is no data showing that the pursuit was conducted that way.
And if you disagree with me, you can contact Prof. Yakir Shoshani for example (professor of physics) to verify the matter.
Miracles!
I did not say that every mathematical matter is necessarily expressed in the physical universe. However, every phenomenon in the physical universe is expressed in mathematical formulas
Israel
You say "when you reach infinity"... and I wrote that it is impossible to reach infinity.... Because there is no end ….. daaaah.
And you say the same thing again 🙂
You are right in distinguishing between a column and a series, although not everyone is careful about the difference. A column does not have a "sum" - a column can converge, like one plus a half and another quarter .... this column converges to two. But (!!!) it is not equal to 2.
I didn't mention or talk about a Taylor series....but, if you already mentioned it - then a Taylor series is by definition an infinite series....In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
You said negligible organs - beware of that 🙂 "neglect" is a concept that depends on the context. Even in a harmonic series, the "last" members are very small, but they are never negligible.
The limit has a definition, and the definition does not talk about the value of the function at the point, but only in the immediate vicinity. This is also what Zeno talked about (the paradox of the arrow).
Israel - the idea in the paradox is to find the error in the way of thinking. Zeno expanded on the ideas of Permendes, who was his teacher. Plato admired Permendes. I don't think it's right to say that these people were idiots, or that they didn't know Hadua. I think they were ahead of their time and overcame a real problem of confusion between math and reality. The accepted explanation for Achilles and the tortoise, with the help of a column sum, is like saying that sine explains Snell's law...
A lesson in Hebrew
Describe a world to you.
Not necessarily the real world but one that is similar to our world but in which time and space are continuous (perhaps it is also similar to our world but we don't know that).
It is an imagined world - not the real world - in which time and space are truly continuous: if we choose a zero point and draw a straight line through it in this world - every real number will have a point that represents it.
The same goes for time: if we choose a certain moment as time 0 then every real number t will have a point in time that it represents.
By the way, the people in this world speak the "B" language.
They call their "space" "Mbarhabbab" and their "time" they call "Zvamban".
The movement in this world is a continuous function from the Zwamban domain to the Mabarbab domain in which for each value of Zbamban there are matched three Kuboobordibinovets of Mabarbab.
A philosopher (Fibilobosovov) who lives there named Zvnobon, says that although everyone notices that Abkhibilevs achieves the Tzvab in the sense that there is a Zvamban t in which the Kuboovovordibinbattev the Merbhabek of Abkhibilbs is the same as that of the Tzvab and in all the later Zvambabenivs (that is - for all T>t ) the Berhabkovo of Abkhibilbs is greater than the Berhabkovo of Well, it's actually impossible.
He justifies his words with the arguments used by "solvers" of the Zenon paradox in our world.
Is he right? After all, we built this world with Zbamban and Mebrhabbab Ravtsifipivib!
If he is not right, then why are the same considerations correct in our world?
Miracles:
"Michael's claim, as I understood it, is that the methodology does indeed describe nature"
Not true.
This is not my claim.
My only argument is that the only way we currently have to describe nature is mathematics, but mathematics is capable of describing many other things as well.
And all this is not at all the main point of my speech.
What I am saying is that the described paradox - like all paradoxes in general - deals with mathematical entities.
You too - all the arguments you try to bring - whether right or wrong - are mathematical arguments and not "natural" arguments.
Your words about the sum of the column are not correct and also the sum of a column is a mathematical entity.
Throughout this discussion I have not changed my mind on any issue and if you say that at a certain moment I claimed something different than what I am claiming now - you misunderstood me at least one of the times.
You are completely wrong in deriving the claim that the universe is not mathematics from the fact that there is something that cannot be calculated. There are mathematical calculations that one does not know how to perform, but that does not make the entities involved in them non-mathematical.
For example - have you heard of polynomials? Did you know that there is no general way to find the roots of a polynomial of degree higher than 4? Are you saying that polynomials are not mathematical entities either?
And does the inability to present something precisely make it non-mathematical? How about root two? Do you know how to write it numerically? When you try to present it numerically aren't you using an approximation?
Miracles:
Yes. You are wrong.
The number has no ambitions. A number is fixed. What tends to 2 is the series of partial sums of the column. The total sum of all column members is 2.
Miracles
This is a book in Hebrew. I went through the books at home, I couldn't find it. But I remember that it was written that the sum was 1 or 1-, and that Leibniz was wrong when he said it was half. In the link I provided, there is no decision in any direction.
So when does the sum of the terms in the series reach 2? Not at infinity?
ב
In English, the definition of series is: A series is, informally speaking, the sum of the terms of a sequence
So apparently literally series means column.
Have you heard about the television series "The Beautiful and the Brave"? The column includes 120 episodes.
So after the semantic tirade, what exactly are you saying? Does my column have no amount? Is it not defined? He can be 77 or i?
Reminds a bit of the discussion on Zeno's paradox.
And who said that the development of Tor Taylor belongs to the field of borders? It is simply a method for calculating the value of a function at a certain point that is difficult to calculate directly. If I remember correctly, this is how M=C^2 or some other Einstein formula is calculated.
What I said is that in calculating a Taylor column, sometimes the last term is the one that gets you closest to the result, as opposed to a column sum such as
...1+1/2+1/4+1/8 where the first terms are the significant contributors to the sum, and the last ones are negligible.
Israel:
Not sure if you are serious or joking.
Anyway:
1) Division by zero is meaningless. Do not divide by zero. It has nothing to do with definitions or fences.
2) The sum of the calculation is not the sum of the infinite column. The sum of the calculation can be at most the sum of a finite number of elements from the infinite column.
3) An infinite column has (or does not have) a sum. An infinite series has (or does not have) a limit.
4) What you wrote down is a column and you didn't arrange the sum of the column is a number. Nothing goes or approaches or arrives. (Even a series doesn't go, doesn't approach, and doesn't arrive, it just exists.)
5) As far as I know an open Taylor column of a function does not belong at all to the issue of limits.
The development yields a column. If the function is defined then the sum of the column is the value of the function. If the column is infinite then its sum is the limit of the series of partial sums.
6) A series can converge to a limit or diverge (no limit). But the calculation of partial sums does not converge and does not diverge, it simply gives a certain result.
Israel
You wrote "+1/2+1/4+1/8 which gets closer and closer to 2 stubbornly, until it reaches infinity."
This is not what is taught…..the definition of infinity does not allow you to reach it.
Israel
The book is Ferraro, Giovanni (June 1999). "The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics". Archive for History of Exact Sciences
The book is published by Springer Link.
Not that I read the book... I must have learned it sometime....
The root of the word "defined" is a fence.
When dividing by 0, there is no limit to the result that can range from minus infinity to infinity.
In our case, the definition is clear: 1 on the right and 0 on the left. The calculation amount is enclosed between these 2 fences.
Therefore you are all wrong.
And you are all right:
http://en.wikipedia.org/wiki/History_of_Grandi's_series
Zvika the maniac stole the book from me (I don't remember the name) in which the answer 1 and 1- appears, and where it says that Leibniz was wrong. Nisim, you said Leibniz before, do you have the book or a link?
And by the same token: usually when an infinite column has a limit, then it aspires to it faithfully. This is what happens in the case of the series...1+1/2+1/4+1/8 which gets closer and closer to 2 stubbornly, until it reaches infinity.
In calculus, the method for calculating a point limit of a persistent and snoozing function is to use a Taylor column. The interesting thing is that with this method sometimes the calculation can stray far from the final limit, and only the last derivative surprisingly yields the correct result.
Israel
You wrote "But as Michael said, this is not a complete picture. Because in defining the problem there is no size or time limit for Achilles steps, and they can aim for zero. Therefore the infinite column converges to a finite size, and Achilles passes the tortoise as we know from experience."
Michael's claim, as I understand it, is that the methodology does describe nature. I think it is not so. In mathematics - the sum of the series is a limit and the limit cannot be reached. Therefore, this is not, in my opinion, a solution to the paradox - and this is also what Michael said in the quote above, even though a predecessor claimed otherwise...
Another example is the 3-body problem - we do not know how to calculate the trajectory analytically, but only with the help of approximations. Still, 3 bodies have a defined trajectory. Perhaps one day an analytical solution will be found (although Poincaré proved not). All I'm saying is there's an open spot here. The universe is not mathematics, at least not today.
Michael
I asked a question - is the column sum to 2? In high school we learned that the sum is not equal to 2, but strives for 2, as close as we want. am I wrong?
Israel:
What everyone tells you is true.
The amount is not specified.
Miracles:
Will you remind me who talked about modesty?
Do you now want to send back to high school not only Israel and me but also you? Wikipedia And all its editors and all the readers who didn't know how to remind them of their mistake?
Israel:
This is not the sum of the infinite column.
This is its series of partial sums.
I do not understand you, friend, what is not clear?
Try to add the numbers. You will see that in each member you add, the sum goes once to 1 and once to 0.
And this is indeed the sum of the series: 1 and 0 alternately.
Good night.
Israel
B is right.
In addition - you did not answer me - is the sum of the column equal to 2? Or is he just getting closer….?
Israel:
The sum of the column is neither 1 nor 0.
At most it can be said that both point 1 and point 0 are accumulation points of the partial sums.
But the series of partial sums does not converge to any limit.
reminder:
Boundary definition:
A certain number is the limit of a series only if starting from a certain term in the series all the other terms are as close as we want to that number.
What is gathering? What is this here, Sanhedrin?
The sum of the series is either 1 or 0. Do you see another option?
Israel
What does 1 or 0 mean? Are you claiming that the series converges to two boundaries?
1 or 0.
Leibniz said half.
Michael / Israel
The column 1 + 1/2 + 1/4 + 1/8 and so on is not equal to 2. Those who don't know this should go back to high school…….
Israel - do you mean Leibnitz?
Israel:
This too is of course not defined, but for other reasons.
I don't know who the famous mathematician is who got it wrong.
What is the sum of the infinite series:
……1-1+1-1+1-1+1-1
And who is the famous mathematician who gave the wrong answer to the question?
Joseph:
There is a series here.
Do you want to claim that the series 10, 11, 11.1, 11.11 and so on is not a series?!
There are actually many (infinite) series, but talk about a specific series and the whole sense of paradox comes from a misunderstanding that this series of events takes place entirely before 11:XNUMX.
incidentally:
Any attempt to connect the current story with questions of continuity is ridiculous because the entire description of the "paradox" is limited to rational numbers and it does not require any continuity at all (all it requires is a misunderstanding)
Israel:
You understood almost everything.
When I adopted in response to which I pointed out the word steps that was mentioned in the responses before - I did not mean steps in the usual sense but what I hoped my predecessor referred to - that is - artificial steps that define the normal and continuous running of Achilles.
That's why I used the word events later.
If someone meant real steps then he got confused in another matter which also has no paradox.
He simply described an unspecified situation because he did not describe how Achilles ran in general but only how he ran 11 seconds ago.
All Achilles does in these steps is to get closer and closer to the meeting point and he does this in infinite steps whose frequency tends to infinity.
Asking what will happen next based on this definition is like asking what the state of a lamp will be (on or off) after two seconds if someone turns it on, turns it off after a second, turns it on again after half a second, turns it off again after a quarter of a second, and so on.
It is simply not defined and there is no paradox here.
Miracles:
I suggest you equip yourself with some modesty.
You make statements that are generally rejected by the scientific, mathematical, philosophical and physical community (except for a few philosophers who don't understand) and you try to create a presentation as if only you understand what is going on.
There is no "researcher" who thinks otherwise because it is not a subject of research. It's just a matter of thought.
You can talk as much as you want about the differences between mathematics and physics and it will not change the fact that everything you have called "physics" up to now (and everything you will call that in the future) is mathematics.
The only thing that makes a certain mathematical structure part of our physical understanding is the connection we recognize between it and reality, but it remains a mathematical structure.
Paradox is also a logical/mathematical concept.
Yossi Simon
I have no argument about the math. But it is a mistake to think that mathematics and the universe are equivalent. Not everything that exists in the world of mathematics exists in the universe.
I understand that you are misinterpreting the paradox. Zenon did not try to prove that there is no motion - he was far from stupid.
Those who claim that a mathematical limit is the solution are fooling themselves - by definition, the limit cannot be reached, but you can get as close as you want. I gave 2 links above - worth reading them.
Hello miracles!
1. The world of mathematics (or the group of mathematics which is an infinite group) does not depend on the universe, man, or any other "exist" and is a completely abstract world. (The Pythagorean Theorem always holds whether the person discovered it or not, the equality years and more years is equal to four, not even Putin can change).
Physics uses it to simplify and minimize and unify its laws. It doesn't matter what geometry it uses. And therefore Zenon's paradox is also actually a mathematical paradox. That is, in solving an equation with two variables there is a paradox. You can expand and say that any sequence of any kind like time for example contains a paradox and it is apparent.
In the story of Achilles, they try to prove that it is not possible to go through a sequence of infinite points while basing on the passage of infinite points since every smallest step of the tortoise/Achilles contains infinite points. (Achilles did not reduce his steps as some commentators believe).
2. The world of science contains endless disputes between smart and humble people, so it must be assumed that some of them are sometimes wrong.
3. At the time the paradox was born in order to clarify the concept of "movement" whether there is movement at all. And that's perfectly fine because every little detail is not something to be taken for granted.
I will try again
Israel
The article refers to the Super Task problem which is exactly the problem in Paradox.
You are also welcome to read here - http://www.iep.utm.edu/zeno-par/#SH5b
And here - http://www.physicsforums.com/showthread.php?t=613003
stuck…………………….
Israel
The article refers to the Super Task problem which is exactly the problem in Paradox.
You are also welcome to read here - http://www.iep.utm.edu/zeno-par/#SH5b
And here - http://www.physicsforums.com/showthread.php?t=613003
Miracles
I asked you for a link that explains the problem you claim exists in Zenon's paradox. You referred to a link where the problem is not even mentioned.
So either you turn, or you will see your solution, or you will be deaf forever!
😀
Chastity Alek..
I suggest you (no names) to equip yourself with some modesty. There are fundamental differences between the physical world and the mathematical world.
I won't go into topics such as the assumption of continuity here, but people like Bergson, Thompson Whitehead and Pfefferman (wise and humble....). But, in accepting the standard solution you have to give up all kinds of intuitions that we have. For example - that it is impossible to decompose a sequence into a collection of points.
I want to emphasize - all I am saying is that according to many researchers, the standard solution does not solve the paradox. I have my own personal solution that I have already detailed before, but this is just my opinion...
So everything works out. I understood from
https://www.hayadan.org.il/galaxy-from-beginning-of-the-universe-2510130/comment-page-3/#comment-453410
There is some version where each step is ten times smaller than the previous one. In such a case it is indeed possible that Achilles will not get the tortoise.
Greetings to Arnon Aver..
Maybe it's better to leave a pass?
Michael!
There is no series here but two continuous (or straight) functions that are not parallel. The story about a series is designed to create the illusion of paradox.
Until the meeting point, the distance can be divided into an infinite number of points, so what?
And by the way - the name of my good friend Arnon Avron is Arnon Avron
Israel:
It's very simple!
First of all, the problem does not talk about the size of the step at all and as far as the paradox is concerned, they could also travel by bicycle so this whole story about smaller and smaller steps is irrelevant.
When I talked about an infinite series of events, I meant the event of "Achilles' inventions at one point or another" (and don't start me now with "Achilles is not a point body because I will immediately set a point on Achilles whose position I will identify with Achilles' position).
This is an event that Achilles does not even have to think about: Achilles runs forward and the events happen by themselves.
At a certain point all these events pass and Achilles is in the part of the route and in the period of time that Zeno did not refer to at all.
On second thought - reverse reaction. It is clear that if Achilles steps tend to zero, the sum of the column can also tend to a finite size, and it will get stuck in place. Apparently the problem is really a bit more complex.
Michael, could you explain how if, as you say, "each one is ten times smaller than the previous one" then why won't Achilles get stuck after a certain distance that is not necessarily greater than that traveled by the tortoise? After all, such a column does converge on the 11th and the ninth as you wrote, so how will it pass the turtle?
Besides, where in the original problem does it say that each step is ten times smaller than the previous one? In my understanding, the original paradox does not have this condition.
ב
All of Hadoa, which is a mathematical field, came from speed and acceleration calculations, which are a physical field.
The problem you raise "Achilles at some point reaches a step size that is smaller than the diameter of an atom" exists in every derivative of a function, where the denominator tends to zero but does not reach it.
And despite the contradiction that supposedly exists in infinity, since we finally put the zero in the denominator, the results obtained - mathematically and physically - are correct.
In Arnon Evron's book "Godel Laws and the Problem of the Basics of Mathematics":
http://simania.co.il/bookdetails.php?item_id=16951
The problem of the chicken's knees is raised, on which the entire field of calculus rests. If I remember correctly, it was Lohbittel who was able to establish the calculus as a truth in mathematics.
So you and Miracles are right when you claim a supposed contradiction in the real world, where Achilles has to take an infinite number of steps to reach the tortoise. Because if we assume that each step takes time that does not tend to zero, then in practice Achilles will never reach the tortoise.
But as Michael said, this is not a complete picture. Because in defining the problem there is no size or time limit for Achilles steps, and they can aim for zero. Therefore the infinite column converges to a finite size, and Achilles passes the tortoise as we know from experience.
B:
The way you describe as physical is also mathematical.
By the way, this way is not at all what is described in the problem because they do not talk about the size of the steps there at all.
In any case, as I explained, go back and explain, there are two ways of thinking about the problem, both of them are mathematical and in one of them there is an error of drawing a conclusion that is clearly not justified, not mathematically, not physically or in any other way
Miracles:
I didn't say there were no other opinions.
After all, there are also other opinions about God and the Holocaust.
I only said that the other opinions are wrong and unlike the other opinion holders who only express their opinion and say that mathematics does not capture everything but do not point to anything that it does not capture, I wrote only well-founded and reasoned things
Achilles and the Tortoise: Mathematics and Physics:
The problem can be approached in two ways:
1) Mathematically:
Achilles moves forward and closes the distance between himself and the tortoise.
The form in which he does it is not important. He can also ride a bike. for a ride in the car. or fly by plane.
We refer only to the distance between Achilles and the tortoise.
The solution to this problem belongs to the theory of limits and is a convergent column.
2) Physically:
Achilles ran. He takes steps with his feet. Each step is smaller than the one before it.
The solution to this problem is different because Achilles at some point reaches a step size that is smaller than the diameter of an atom.
Achilles cannot continue to decrease his steps as required in the description of the problem.
Michael
Respected people write doctorates on the subject. Not everyone is stupid….. To be fair Wikipedia has other opinions too.
Zeno's paradox
This paradox is built on an assumption that they want to disprove.
It is "proven" that Achilles cannot pass 10 meters. So for the same reason both the Tortoise and Achilles cannot move any distance not even one millimeter. But the claim is based on the fact that Achilles did advance a certain distance. And the apparent conclusion is not that Achilles will not catch the tortoise, but that a state of motion is impossible (and more than that).
The departure from the paradox - it is necessary to refer to time and indeed within a certain time period Achilles will not reach the tortoise.
Why is it similar?
Let's look at the following two graphs:
1. X=5+Y
2. X=2Y
At point 0:0 the height of the first graph is 5 and that of the second 0.
When the second graph reaches a height of 5 (when X=2.5) the first graph will rise a little. And so on.
So then graph 2 will never reach (intersect) graph 1. (A world where X=5, the height of the two graphs is 10)
And here there is no involvement of physics and matter. But simply a problem of phrasing logic and thinking.
Conclusion: that everyone should doubt what he is convinced of and do not rush to criticize others.
From "The Trial of the Second Law" (Where is Shmulik?).
That's why everyone was so excited when little i joined the family, and held a religious and proper feast in his honor, intended, they promised, for the whole world of numbers. Who wasn't there? Every fat man of mathematics, every duke and every count, and everyone who is a little something. Endless columns, which had gathered especially for the occasion, marched in total. Entertainment stages were set up for series.
Let's summarize the paradox as follows:
Philosophically - Achilles will not catch the tortoise.
Which is obvious, because according to the field of philosophy also known as "Who am I, what am I" Achilles is actually the tortoise and the tortoise is actually Zeno.
And therefore Achilles will not even achieve himself, at most he will achieve Hector.
Physically - he will pass the turtle lightly and leave it behind.
On the contrary, miracles:
It is precisely you who do not understand the essence of the paradox while Israel, I and Wikipedia do.
It is clear that apart from a mathematical problem we have nothing and all our thoughts on the subject deal with various mathematical models and nothing else.
And that it is not clear whether there is something infinite in the world is not at all related to the paradox that discusses a hypothetical world where there are infinite things.
To start talking about a world other than the one discussed by the paradox is simply evasion, on the one hand, and proof of a misunderstanding, on the other hand
Israel
I agree with you. But neither you nor Michael understand what the paradox is talking about.
It is clear that mathematically there is no paradox here, and it is clear that Achilles gets the tortoise.. What is not clear is the meaning of the word infinity. By definition - you cannot finish a series of infinite actions. To say "it is the sum of an infinite column, and this sum is well defined", according to certain philosophers, does not solve the paradox. There is an interesting concept in philosophy called supertask. At least it's interesting to me...
It is not at all clear if there is something infinite in the world. And that goes both ways. It is clear that time is not infinite in both directions (it may be a ray, but it cannot be straight (no beginning and no end)). Nor is it clear that time is infinitely divisible. For example - does a photon reach speed c at time 0?
In short - Zenon tried to draw the conclusion from the fact that there are tenses of the form 10, 11, 11.1, 11.11, 11.111, 11.1111 and so on that the 11th tense does not exist or cannot be reached
Israel:
The wording you referred to was probably a bit vague but it meant steps that are each ten times smaller than the previous one.
Throughout this series of steps Achilles will not really catch the tortoise.
The mistake is assuming that these steps are the be-all and end-all.
We have here a series of events that does not have a final event but there are events after it and that is what Zenon forgets.
Miracles
I read the link. I have not seen any explanation, mathematical physical or philosophical, for Zeno's paradox. Did I miss something again?
Saturday, boy, football.
"At most it can be argued that Achilles will not reach the tortoise in a finite number of steps. (Steps like those described in the presentation of the problem)".
What are the steps described in the problem presentation?
Surely Achilles would catch up with the tortoise in a finite number of steps, and not a very large one either.
Give me the fur of the tortoise, speeds and the tortoise and Achilles, and the size of his stride. Tells you in how many steps he will achieve it.
And by the way - I did not say that there are no philosophers who do not understand mathematics or the connection between it and physics.
For everything that is true, you will also find people who say that it is not true. It doesn't impress me at all and it's completely clear to me that the paradox has been resolved
Miracles:
I see you don't understand what I keep trying to explain:
Apart from the mathematical equations we have nothing. We understand the world through models. Achilles and the tortoise are also models - mathematical entities - and the whole paradox is in the realm of logic and mathematics.
What you said about the simpler solution shows that you also did not understand the meaning of the word paradox.
A paradox is created when there is a contradiction between the conclusions of two ways of thinking that seem legitimate to us.
In the case of Achilles and the tortoise - these two ways are the solution of the linear equations on the one hand and Zenon's consideration on the other.
The solution to the paradox is not obtained by arbitrarily deciding to choose one way without showing what is wrong with the other. It's not a solution to a paradox but an ignoring of a paradox
Israel
Read the link I gave Michael.
Mathematically - 2 equations in two vanishes solves the problem. An infinite column is also a mathematical solution.
Physically - an infinite column has no meaning in physics.
The question is in philosophy. No one claims that Achilles doesn't beat the tortoise. If the column is infinite then it has no end…..it has infinite terms.
B:
Still you brought up Achilles and the tortoise as an example of something.
An example of why - if so?
Be that as it may, a correct mathematical understanding of the matter shows that there is no paradox here.
Your understanding of the big bang theory as if it claims that there was no time before is also inaccurate.
Actually we do not know this and there are also speculations about the possibility that there was a time before. The assumption that there was no previous time is not part of the theory.
In Wikipedia you will find in the chapter on the big Bang The following sentence:
"Our understanding of the Universe back to very early times suggests that there is a past horizon, although in practice our view is also limited by the opacity of the Universe at early times. So our view cannot extend further backward in time, although the horizon recedes in space. If the expansion of the Universe continues to accelerate, there is a future horizon as well.[36]"
The theory does claim that the familiar structure of "time-space" was created then, but it does not really contradict the existence of a previous time.
I remember that in a meeting I had many years ago with Yakir Aharonov, I asked him "How is it that the cosmologists claim that the big bang in which they claim time was created is the result of a quantum fluctuation when the whole term of fluctuation is defined as a change of something over time?" He told me that they really think that there may have been a "kind of" time before, but apparently this time was something different from the time we are familiar with.
Michael
For example:
https://docs.google.com/document/edit?id=1D86gx8yjYXVBTBNEbtPCe4VEQdso6bwAA9ot8sYEY4M&hl=en&pli=1
Michael
A nice math exercise does not solve the paradox. There is a much simpler solution than the sum of an infinite column - solving the system of equations of two variables. Why bother with an endless column??
You have the right to think that the paradox is resolved, but in philosophy circles the issue is not closed.
ב
Meztomar "I don't know anything about the way they came to the conclusion that the universe started with the big bang.
It just seems to me that there is an internal contradiction."
If all galaxies are moving away from each other, doesn't it make sense that they were once closest? that they all came from the same point?
It is even possible to calculate when they were at the same singular point: about 13.7 billion years ago.
It is also perfectly consistent with the predictions of general relativity.
Miracles
Do you have a link that expands on the subject and explains why converging columns are not the solution?
And I don't mean in the philosophical realm. Just physics and math.
Michael:
Of course Achilles will get the tortoise.
And he will even get it very quickly.
I am not claiming that Achilles will "never" get the tortoise.
At most it can be argued that Achilles will not reach the tortoise in a finite number of steps (such steps as described in the presentation of the problem).
Nor am I claiming that time and space are not continuous.
It is precisely the claim that they are continuous that seems correct to me.
All I meant to say is that a particular mathematical description is not necessarily the correct description of the physical state and therefore if a particular mathematical description produces a contradiction another mathematical description must be formulated.
I will admit and I will not be ashamed that I know nothing about the way they reached the conclusion that the universe began with the big bang.
It just seems to me that there is an internal contradiction.
I will be very happy if I find out that it just seems to me and there is no internal contradiction.
Miracles:
Speak for yourself.
I know how to solve this paradox and in my opinion converging columns is definitely the solution.
You're welcome Read about it on Wikipedia as well.
B:
I am saying a very simple thing: anyone who thinks there is a paradox here does not understand that a very simple logical error was made here of checking things that necessarily happen before a known time and deducing a statement like "never" (which talks about times after that time) from that check.
You say that Achilles is a physical being and I say that he is a mathematical being and then you say "he is what I said".
It's really a paradox.
A paradox is a situation where two ways of thinking that seem legitimate to us give contradictory results... But why am I repeating the article - I explained everything there and I should have just sent you to read again - and this time carefully.
All our thinking about the physical world is done through models which are mathematical entities.
The quantum model is also a mathematical model.
When they come and describe to you a paradox based on a classical model of the world - the statement "this is not the true model of the world" is not relevant at all. The question is a pure logical question and the attempt to conclude from the Achilles and the Tortoise paradox that time and space are not continuous is simply ridiculous.
By the way - of course scientists know this and no one claims today that time and/or space are quantum.
There is indeed such a possibility and no one rules it out, but everyone understands that the story of Achilles and the tortoise cannot draw this conclusion, so today it is an open question.
You are welcome, for example, to read on Wikipedia about Plank length which is occasionally offered as the minimum possible distance and understand that to date no justification has been found for this.
Not for nothing is it written there: "There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research.”
Israel
Don't get into the subject of Achilles and the tortoise...we don't know how to solve this paradox today. And "converging columns" is not the solution...
I claim:
1) Achilles will indeed get the tortoise.
2) A physical approach to the problem of Achilles and the tortoise is significantly and fundamentally different from a mathematical approach to a convergent column.
The mathematical solution that gives the limit to which the column converges lacks a physical description of the real process.
In the real process, factors such as:
a) The size of the smallest step that Achilles can take.
b) The greatest rate of steps that Achilles can take.
If, for example, Achilles reaches the size of the smallest step he is able to take and still has a large distance left to the turtle, then he will not be able to persist in reducing his steps and therefore there will be an exception to the mathematical description of the problem. That is, the physical process reaches a limit where the mathematical description no longer fits.
ב
what are you claiming That Achilles won't get the tortoise for some physical reason, classical or quantum? Could you give a specific example, i.e. the distance in meters of the turtle and speeds in m/s at which Achilles does not catch the turtle?
Have you heard of converging columns?
He said:
If
taking a physical problem and turning it into a mathematical problem, while ignoring physical factors that are an integral part of the problem,
Then
get a mathematical solution.
But this is not always the physical solution.
If
The mathematical solution of the equations of motion of classical mechanics would describe the physical reality
Then
Quantum mechanics was not needed.
B:
It turns out that you have more to learn - and not only in physics.
The story of Achilles and the tortoise is only paradoxical for those who do not understand mathematics and logic.
You may not know, but Achilles and the tortoise do not exist in reality, you only know them from the story about Achilles and the tortoise! Therefore they are not physical objects. On the contrary! They are mathematical objects!
וThe solution to the paradox - also from the field of mathematics.
Michael:
a) There are cases where equations are solved in a certain way but they do not describe reality.
Example:
Achilles and the tortoise.
Despite the theory of limits. Nevertheless, the problem of Achilles and the tortoise is not solved and this is because Achilles is not a mathematical being.
Achilles is a biological creature.
A biological creature has limitations:
1) The size of the steps cannot be as small as we want.
2) The number of steps per second cannot be as large as we wish.
That is:
The very presentation of the matter is fundamentally wrong.
We solved a math problem but we didn't solve the problem presented.
About the presented problem we can only say that its formulation is wrong and the situation described in it is impossible.
b) I thank you for the reference and the attempt to explain.
Piqued my curiosity. It is clear to me that I must study the subject well in order to understand what it is about.
Thank you!
If the mass and gravity appeared immediately after the bang, then all the mass and energy were in the Schwarzschild radius.
How did all the mass and energy not collapse into a black hole?
B:
I said you need to study and I don't go back on my words.
As long as you don't know everything there is to know, you will keep asking and that simply cannot, technically, happen here on the site.
Physics has already taught us that time is not exactly what we thought.
The theory of relativity first did this when it revealed to us that time is relative.
According to the theory of relativity - massless particles necessarily move at the speed of light and what moves at the speed of light has no time (and not because it is in a hurry).
For photons, for example, not much time has passed since they were first created.
If at some point the universe consists only of massless particles it has no time.
At least this is the theory proposed by Penrose (the mathematician and theoretical physicist who discovered even before Schachtmann why quasi-crystals are possible).
It can be argued that when there is no time there is no justification for using the term "before the present universe" and this is justified in many ways but in the absence of a term that describes what happens when there is no time this term is used.
As mentioned - in terms of photons, for example, all universes exist at the same time.
You will not dictate to the ants what to do when the balloon inflates.
It is a fact that when a balloon inflates the ants do not inflate with it.
Although a force transmitted from the balloon to their legs is constantly applied to them and tries to "stretch" them, the internal forces holding them overcome this force and they do not stretch.
The same thing probably happens to galaxy clusters: the gravitational force acting in the content overcomes the expansion of space.
On the other hand - the distance between the clusters increases because the space - unlike the masses held together by gravity - has no constraint that hinders its expansion.
As mentioned: I will not be able to give you in the comments here on the website everything that you do not bother to go to study in an orderly manner, so I will stop here, especially in light of the fact that just now my grandchildren came to visit.
I have a problem with the article. I'll explain what doesn't work for me:
The universe has existed 13.8 billion years, the galaxy we see today is from 13.1 billion years ago.
All galaxies were formed as a result of the big bang and began to spread and move rapidly in all directions.
Think of a balloon that inflates.
How is it possible that light from 13 billion years ago is reaching us now, since it should have passed through our galaxy a long time ago
Michael:
1)
A) According to your words, I understand that before the zero time of our universe there was another universe.
But how does the existence of a zero point in time work out when it is clear that before zero there was another universe?
b) If there is a transition between the universes then time also has an end.
c) If there is a series of universes then there is a first and there is a second and there is a third and so on.
That is, there is a time that does not depend on the existence of the universes.
2)
If the dimensions inflate then both the balloon inflates and the ants inflate.
If you mark the years on the balloon, you will see that the distance between the years remains even if the balloon inflates.
That is, measuring the distance between the ports according to the marked years will yield the same results whether the balloon is inflated or not.
Therefore the speed of the ants when measured according to the years marked on the balloon will remain the same even if the balloon is inflated.
That means the speed of the galaxies in meters per second will not change even if the universe expands. Therefore, it is not possible for galaxies to move away from each other at a speed that exceeds the speed of light.
ב:
Zero time works out because in our universe there was no time before.
I mention that the theory of a series of universes is currently only one of many speculations.
According to this speculation there is no connection between the time of the previous universe and the time of the current universe because in the transition between them time lost its meaning.
This is related to the fact that by combining the insights of relativity and quantum theory it can be concluded that if there are no particles with rest mass then there is no time either and this theory suggests that at a certain point in the transition between universes only particles that have no rest mass remain and time is lost.
I know it sounds like Chinese to most people and I already said that to fully explain it Roger Penrose saw fit to write a whole book (which is not easy to read).
I mentioned this book in the link I gave in a previous answer.
Kfir:
You asked about books and the above links are links to books on the subject sold by Amazon.
ב:
Don't you see for yourself the difference between the speed of the ant on the surface of the balloon and the speed of it moving away from another ant that results from the inflation of the balloon? After all, this distancing can happen even without the ant moving its legs and without doing anything to move! She can even move her legs towards another ant and yet move away from it due to the balloon inflation!
Kfir:
Regarding the topology that resembles the surface of a sphere, you are invited to expand your understanding of the subject with the help of the book "Poincaré's Conjecture"
http://www.amazon.com/The-Poincare-Conjecture-Search-Universe/dp/080271532X/ref=sr_1_1?ie=UTF8&qid=1382736560&sr=8-1&keywords=poincare+conjecture
The idea of the similitude before a sphere was reached in many ways, but in my opinion, the best evidence for the matter is the increase in the rate at which the stars recede as a function of distance and the constant acceleration of this rate.
In the link I provided regarding the observable universe there is a certain reference to the matter and this reference also links to others.
http://en.wikipedia.org/wiki/Observable_universe
Michael:
I do not understand the field, I understood the metaphor of the surface of the ball, could you expand on it? How did they find it out?
And in addition to the theory of the parallel universes, and the universes that arise from a previous location, and a new universe emerges from our universe. I'm not that familiar with this theory, could you refer me to the literature that deals with these theories, and recommend introductory books to the field?
Thanks
Nadav, one of the objections of respected astronomers like Fred Hoyle to the big bang theory is that it brings God back in the back door. But scientists get along just fine with Big Bang physics without involving God.
Dear Avi Blizovsky, regarding your response: I suggest some modesty.
Some people think they are better than others, with them "it doesn't happen", they are rational and enlightened and everyone else who doesn't think like them... they don't.
I don't know if this is news to you - but everyone reads what they want to read, it's an inherent human trait
At one level or another with everyone.
And if you have something smart to say about my existence. I will be happy to hear.
Michael:
Science is less interesting when you don't respond to laymen. Mainly for those who have just learned how to write astrophysics without errors, and have already come to the conclusion that the Big Bang theory contradicts itself, the Torah contradicts every scientific finding or to contrast it already reported it 2000 years ago, but only now do we know what was meant. I always thought that if God said "and let there be light" it means that he invented the darkness before that.
For all people of the Torah, the Tanakh, the Talmud, the New Testament, the Church of Scientology and Zen: there are enough websites that advocate the approach that the earth is the center of the universe and has only existed for 5000 years. It felt comfortable to respond there with the nonsense and say that everything was already written in advance, and only the stupid scientists who do not know how to interpret the Torah did not understand that every research solution that thousands of scientists around the world have been working on for decades, was already written on page X in book Y.
On that occasion, you are invited to give solutions from the Torah to scientific questions such as: how many universes existed before the present one, how to control the weather, how to create an atmosphere on our moon and what is the chromosome that controls telekinesis!
1) How does the concept of the zero of time work out when it is clear that there was a zero before it in the previous instance?
2) If there is a difference between the speed of the ants relative to the balloon and the speed of their distance from each other, then what would you define as speed?
Does speed have two different definitions?
B:
It was clear to me that you would not be clear.
That's why I advised you to study before you make statements.
The universe you are talking about - if it includes all the shows - is not the universe scientists are talking about when they are dealing with the possibility of an infinite series of universes. In any case - it is not clear what your question is in this regard. Is there a question here or is it simply a demand from the scientists to align themselves with you in their language?
In relation to the expansion of space, you can imagine ants walking on a balloon.
The speed of movement of the ants is limited by their physiology but this does not prevent them from moving away from each other faster than they walk if while they walk they inflate the balloon.
ב
There is no point in me writing the same comment again. You form one sentence from science, don't understand it, but use it to dismiss the rest of science.
1) It is not clear what negative energy is.
2) The universe I'm talking about includes all the shows. One show is not the universe.
If there was a previous instance then there was also a time before the zero point of the current instance's time.
It is also not clear what the expansion of space is.
And if we assume that the space expands then the dimensions of the space also expand along with it and therefore every measurement we make will yield the same results. In particular, measuring the speed of galaxies.
What is speed? Isn't the pace of moving away defined as speed?
Yogev
Do you really think that the story of Genesis is somehow related to reality? There is not a single sentence there that is even close to the truth. In general, the Bible describes the world as they thought then - a flat world that stands in the center, and everything surrounds it.
ב
There is no obstacle for a star to move away from us at a speed higher than the speed of light. As long as space expands, and motion is not the result of force, there is no contradiction here.
I don't understand where you have the audacity to say that the scientists are stupid. Aren't you much more likely to be the fool?
B:
The authors of the books would not have written books if a short and concise explanation was possible. These are serious people and not barbarians.
Nevertheless, while I really sin to the truth, I will summarize for you some of the ideas that appear there:
1. There is also negative energy and in fact the total energy in the universe is zero.
2. The universe is one in an endless series of universes that are built one on the ruins of the previous one and all the energy that was in the first goes to the second. What enables activity in the universe is not the total energy, but the fact that the entropy at the beginning of its existence was very low, and explanations are given for a process that can reconcile the high entropy at the end of the universe's life with the low entropy of the universe that follows it.
In relation to movement at a speed higher than the speed of light - this is essentially the expansion of space itself. The bodies move in space at a speed that does not exceed the speed of light, but the space itself expands and creates a situation where the rate of distance of distant objects within it can exceed the speed of light.
To Michael:
If I didn't value knowledge and learning I wouldn't bother writing here.
But it is not a good method to send someone to study thick books because they raise some question.
It is much better to give a short and concise explanation.
And the question remains:
How is it possible:
is also
1) Matter cannot move faster than light.
and also
2) There are galaxies that move faster than the speed of light.
Ori:
If we perceive light from any star in the universe it does not say anything about what is happening to that star today. It only means that the distance between the position of that star at the moment of light's departure and our position today can be traveled at the speed of light for a period that is shorter than the age of that star.
The speed of light in a vacuum is constant in magnitude and is not affected by any force (its direction may be affected, the frequency of light may be affected, but the magnitude of the speed is not affected).
Science is based on the assumption that the laws of nature do not change (otherwise - what is the point of science at all if all it deals with is an attempt to find out what the laws of nature are? Tomorrow they will change and no longer benefit us!).
I also think that there is no place for nonsense here, so I warn you about the nonsense that people write here.
B:
Hitler also breathed, but I don't condemn breathing because of that.
If you want to claim that you know better than all the scientists without learning anything then I have no reason to argue with you at all.
About one who knows that even after he studies he will understand exactly what he understood before he studied I can only say that he is a child prodigy of the type "who already knew everything he knows today" when he was born.
You don't have to learn anything and you are welcome to continue thinking that all scientists are idiots.
Michael
If we perceive light from a certain point in the universe, this means, according to my understanding, that we are moving away from each other at a speed lower than that of light.
Regarding the determination of distances and speeds. In my opinion 1. The speed of light is affected by forces such as gravity and different materials, solids gases liquids that we do not know or know at all 2. The existence of different fiscal laws in different periods.
History shows that perceptions and theoretical understandings sometimes change beyond recognition.
You have to respect every question and every concept, in science there are many more questions than answers, much more unclear than clear.
Nonsense or nonsense has no place here.
Science searches and searches and in the end it will understand that God created the world and not an explosion, even for an explosion you have to start with something that will cause an explosion and where did the first particle that caused the explosion come from and as far as I know and understand an explosion does not create life or health like we have a sun that shines day and night air for a person to breathe Perfect..one truth and it is the Creator of the world, His name created us in God's image and we are all obliged to thank Him and to thank Him every day always
Michael:
To send people to study halakha for every contradiction they find is a method of the institutional religion.
The problem is that even after studying all the material the contradiction remains.
Did the energy come from nothing?
Did the momentum come out of nowhere?
Is it possible to rely on the assumption that matter cannot move faster than light just to reach the conclusion that there are entire galaxies that move at a higher speed than light.
The contradiction is an internal logical contradiction and even thousands of books will not solve it.
Either you get a certain discount or you reject it.
You can't get anything and vice versa.
Kafir:
Your responses indicate that you are not familiar with the existing cosmological theories.
According to these theories the universe has no end or center.
The image that is commonly used is that of the surface of a sphere that has no edge and no center.
The image is not accurate because the sphere, which is a two-dimensional surface, exists in a space of a higher dimension and in this space it is possible to speak of a center (not of the surface of the sphere but of the sphere itself) while in the space of the universe there is no such additional dimension.
It's not easy to imagine because it's not the kind of thing we can see. The only way to deal with this is by mathematical representation.
By the way - in terms of size - the universe is probably much larger than 13.8 billion light years.
The swelling process that went through, as well as the acceleration of its expansion, resulted in the fact that it contains objects that moved away (and probably still move away) from each other at a speed exceeding the speed of light.
http://en.wikipedia.org/wiki/Observable_universe
Yosef:
If there is no end there is no beginning?
What an interesting philosophical diagnosis?!
Let's look at the period defined by a positive answer to whether the nonsense in your response has already been said by you.
This period has a beginning (the first time you said the nonsense) and it has no end.
B:
What amazes me is that people who have learned nothing about the subject openly claim that they have noticed in the existing theories an internal contradiction that the scientists who worked hard to formulate and specialized in the subject for years did not notice.
You can be sure that everyone (but everyone!) was asked about the problem and quite a few solutions were proposed (but not trivial solutions that can be presented here).
You can find two of the latest references to the subject in the following two books:
http://www.amazon.com/Universe-Nothing-There-Something-Rather/dp/1451624468/ref=sr_1_1?s=books&ie=UTF8&qid=1382689767&sr=1-1&keywords=a+universe+from+nothing
http://www.amazon.com/Cycles-Time-Extraordinary-View-Universe/dp/0307278468/ref=sr_1_1?s=books&ie=UTF8&qid=1382689825&sr=1-1&keywords=cycles+of+time+an+extraordinary+new+view+of+the+universe
Do you doubt that the theory is proven?
interesting!
Do you know that there is no proven theory in science?
After all, the possibility of refutation is at the heart of science - a theory that cannot be disproved is not a scientific theory and as we know - something that has been proven cannot be disproved.
To find out what can be proven, mathematics exists, and if everything could be proven, there would be no science at all.
The big bang theory survives so far because it has many confirmations and not even one refutation.
As we know, an internal contradiction is a type of refutation, so you can understand that those who examined the subject in depth did not find internal contradictions.
There are still many open questions but there are no contradictions or refutations.
If there is no end there is no beginning
The big bang theory is full of contradictions.
I doubt the theory is proven.
An interesting thought experiment that occurred to me, if a person stands (or sits) at one end of the universe, and looks through the most sophisticated and accurate telescope that will ever be built, and let's say that he will be able to see the farthest end of the universe at a distance of about 78-90 billion light years, what is he looking at? actually looking? If the universe today is between 13.8 billion light years, any greater distance will be meaningless and non-existent, if we assume that a telescope picks up light, which moves at the speed of light, then man will have to simulate another 64.2 billion years until he can see the edge of the universe, in "live" Even if this man existed in the first second of the universe, assuming that the universe expanded at the speed of light, even then that man would not be able to see the edge of the universe.
The only time when he will be able to observe the universe, is at no time, and the only "no time" that I know of, is before the beginning of time, before the beginning of the expansion of the universe, when it was concentrated in one singularity, (assuming that our universe is the only universe, in this dimension , by its very definition) that person will be able to see the edge of the universe when he is concentrated in one point, but when the entire universe is concentrated in one point, there will be no end..
I would love to hear what you think about it.
Something here doesn't add up to me - if light takes 13.1 billion years to reach us from this galaxy, and the universe is 13.8 billion years old, and we know that the universe is expanding, and it started at a singularity. So the light that reaches us now is from the location where the galaxy was at the beginning of the universe and not from where it is today, that is, the center of the universe. And the distance from the center of the universe to the Milky Way galaxy is 26,000 light years, which is pretty close in terms of the universe.
The question is, what are we really seeing? And where do we see it in relation to the center of the universe and the younger galaxies? Is it possible that a much closer galaxy will appear to us more distant than the farthest (observed) galaxy in the universe, just because its light takes less time to reach us?
Another interesting point, if the size of the universe is between 78-90 billion light years, that means that if this galaxy still exists, we should see its light in 13.1 billion years. But since the universe is expanding at an ever-increasing speed, I wonder how long it will be before we can (theoretically) see it as it is today.
And this is without mentioning the effect of gravity on the light we see, on the curvature of the time dimension.
If anyone can enlighten me on this issue?
What is amazing is that the internal contradiction is not noticed:
The science of physics deals with conservation laws.
Conservation of energy, momentum, etc.
Whereas the Big Bang speaks of the creation of nothing, which is in contradiction to the laws of conservation.
Where was all the energy before the universe was created?
Yogev, you probably read what you want to read. Not what it really says.
I'm interested in one thing about the whole story. After several hundred years of a very skeptical science that completely cut itself off from religious traditions and the stories of the Bible. The best story about the creation of the universe is what appears in chapter XNUMX of Genesis - Let there be light, and let there be light.
Could it be that humans are so limited that we have no way to think about anything else, or is this simply the truth
About the creation of the universe, and if so, how did the ancients know this?