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I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on t

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since I never could make that jump, it all sounds nonsense to me. I have a similar (and I believe to be equivalent) problems with infinitesimal as well. How does arbitrarily small but non-zero become infinitesimal? Since I have problem with infinitesimal, I find the differentiation of real numbers and rational numbers equally non-sensical. So mathematician, take a pause, could you explain what is countable infinity? Since you never can finish counting (all the natural numbers), how does it make the set countable? What do we mean exactly by countable here? Wikipedia refers to the idea of one-to-one correspondence. But since you can never exhaust the correspondence, what do we mean by one-to-one? Give me any unique real, I'll give you a unique natural number, and we can go on forever, so how does that not count as one-to-one correspondence? Unlike mathematicians, physicists are fine with unresolved :) PS: I guess countable can be defined as there is a definite way of ordering the set, which is true for natural numbers but questionable for real numbers. I still don't see how the ordering connects to the size of infinity and one to one correspondence. Even for the set of real numbers, I can have an algorithm continuously generate random numbers (discarding re-occurring ones so it will be a unique sequence) and prove there is an order of the set (non-exhaustively defined, same as the set of natural number). The ordering may not be describable though. But non-describable ordering is still an ordering, right? Just as a real number that cannot be exhaustively described is still a number. I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity.

cvoss 10y ago
I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot win this game. Yes, the counting process is non-terminating. But every item gets counted after only finite time.
- hzhou321 10y ago
  Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you? Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?
  - zeeboo 10y ago
    If I start at 0 and successively add 1, do you agree that I eventually hit any positive integer you could pick after a finite number of steps? Does that not prove to you that I hit every positive integers? Which one do I not hit?
    - hzhou321 10y ago
      You will only eventually hit any given integer after finite steps. It does not prove you will hit every positive integer. You'll miss those that one never can finish giving you -- for example, I'll start with digit 1, and I'll infinitely adding 1 behind it (never finishing). It is an infinite natural number that you can't hit within any finite number of steps.
      - cvoss 10y ago
        The game itself is certainly not the proof that is required. The proof in question is a proof that the game cannot be won by the adversary. By no means do you need to play all possible rounds of the game to conclude this.
        hzhou321 10y ago
        The discussion really helps me understand the problem. Real numbers are infinity in disguise. Because infinity is not really defined, real numbers are not really defined. Just saying infinity is not finite does not make much sense (in the sense of adding anything helpful). Any real number that you can finitely describe can be included in a finitely described counting scheme. Let's use the counting schemes of rational numbers, I'll hit arbitrary numbers to arbitrary precisions. Taking a limit, it is not clear that I miss any numbers (including pi). To defeat this scheme, the adversary has to keep adding digits to his real number, as well as keep shrinking his allowed precision infinitely. Now both the number (that is being infinitely being described) and my counting process is infinite, we didn't and can't prove anything. There is simply no conclusions to be drawn.
jonreem 10y ago
A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it. So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function. We can extend this and say that the set of even natural numbers is countably infinite because it has a mapping function of x => x / 2. The same is true for all integers (natural numbers + negative numbers): x => if (x < 0) { x * -1 * 2 } else if (x == 0) { 0 } else { x * 2 + 1 }, i.e. if it's negative map it to an even number and if it is positive map it to an odd number, if it's zero map it to itself. You can even write a function that maps all rational numbers to the natural numbers, since each rational number can be written as a fraction of two integers. (Figuring out the function is a fun exercise but it is also easy to google) However, you can't write a function that maps any real number to a natural number. The easiest to understand proof of this is Cantor's Diagonal Argument[0], which is a proof by contradiction that shows that any attempted function must exclude some real numbers. Therefore, the real numbers are not countably infinite, and we call them uncountably infinite. EDIT: In response to your edit, Cantor's Diagonal Argument basically shows that for any given function (and you have to define the function completely ahead of time - that's key) I can give you a real number that is not included in the domain of your function. [0]: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- hzhou321 10y ago
  Why a pre-definable function matters here? We have allowed a real number (that cannot be exhaustively described) in, so why a real function (that you cannot exhaustively define) has to be excluded? Isn't it unfair that I am only allowed to use a finitely definable function while you can choose a non-pre-finitely-describable real number? Isn't that a loop proof -- that the set of real numbers is uncountable simply because it is defined to be uncountable, and it is larger infinity than that of natural numbers simply it is defined to be bigger (as what bigger means in the context of infinity is otherwise undefined).
  - zeeboo 10y ago
drdeca 10y ago
> Since you never can finish counting Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer. The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something. Also, the specification can't depend on things like, what reals have been given as input "previously". Say that in this game, you have two obligations: 1: for any n,m I give, you must tell me what the mth binary digit of the real number associated with n is, in the mapping you are considering (or, if there is no real number associated with that natural number.). 2: For any finite collection of the binary digits of the real number I am choosing, you must tell me the first natural number such that the corresponding real number matches all the digits I specified. With these rules, I can choose my real number (specifying more and more digits) such that for any natural number, I will be able to show that my real number doesn't correspond to that natural number, or any lower one. Therefore, there is no natural number that corresponds to my number in the matching system you are providing. (To win, I repeat this procedure: Ask what the first natural not ruled out already by my specification of my number is. (Call that n) Ask what the nth digits of the real associated with n is. Inform you that the nth digit of my number turns out to have the other value for its nth digit, so no natural less than or equal to n corresponds to my number. Repeat. This will only ever tell you more about my number, and will not result in me changing my mind about any of the digits of my number, yet there is no natural number which won't ever be ruled out as potentially corresponding to my number. Therefore, no natural corresponds to my number in your system. So I win.)
- hzhou321 10y ago
  Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?
  - drdeca 10y ago
hardboiled 10y ago
Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell's 'Primer of Infinitesimal Analysis' has good details. The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes. Calculus can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve. Also you don't differentiate numbers (real or rational), you can only differentiate functions. Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.
- hzhou321 10y ago
  Seems I agree with you (or you agree with mine) :) I don't have problem with calculus -- or I wouldn't be able to do physics. I am having problem with calculus based on infinity. > Also the idea of an actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent. Well said. > Also you don't differentiate numbers (real or rational), you can only differentiate functions. Of course I meant distinction.
esoterica 10y ago
> Since you never can finish counting You seem to have serious trouble wrapping your head around the concept of words having different meanings in a layman context and a technical context. Just because you can't "count" (in the counting-out-loud-on-your-fingers sense) the naturals doesn't mean they aren't countable in the rigorous, set-theoretic sense. Your entire argument basically boils down to "this word means something different from what I'm used to it meaning in daily conversation". > I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. How narcissistic do you have to be to assume that an entire academic discipline must be wrong just because you can't personally grok it? > I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity. It isn't hand-waved at all. Just because you don't understand it doesn't mean it's not rigorous. Go read a set theory textbook or take a MOOC or something.
- jacobolus 10y ago
  I downvoted and flagged your comment. Please don’t bring gratuitous personal attacks here.
raattgift 10y ago
I'll try to physicalize countability. Start counting the naturals: 1, 2, 3, ... At future timelike infinity you'll reach infinity. Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. 0 0.0........ At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real. This survives across changes of positional counting systems, and almost certainly survives arbitrary choices of non-lossy notation, as long as you start with a finite representation of 0.
- stymaar 10y ago
  > I'll try to physicalize countability. > Start counting the naturals: 1, 2, 3, ... > At future timelike infinity you'll reach infinity. > Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. > 0 0.0........ > At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. > In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real. The same reasoning applies for rationals, yet they can still be counted. The definition of «can be counted» means there is a bijection between your set and the naturals. Such kind of bijection can easily be created for the rationals[1] and Cantor's diagonal argument[2] shows that you can't create such bijection for reals. There is nothing really intuitive about this concept, but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. [1] https://en.wikipedia.org/wiki/Pairing_function#/media/File:Diagonal_argument.svg https://en.wikipedia.org/wiki/Pairing_function#/media/File:D... [2] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
  - hzhou321 10y ago
    > but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. It is only straightforward once you accepted infinity and all other definitions/description based on infinity. If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical. >> Start counting the naturals: 1, 2, 3, ... >> At future timelike infinity you'll reach infinity. This highlights the flaw. You reach infinity with infinity. Nothing is really being said about infinity. But somehow if you accepted the understanding infinity here, the rest of thesis such as cantor's diagonal argument may seem to be natural, except you forget that you really didn't know what infinity is. All property of infinity cannot be finitely described. So anything about infinity is built on top of infinity. Turtle all the way down (or up), and we don't really know what it is.

cvoss 10y ago

I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot win this game. Yes, the counting process is non-terminating. But every item gets counted after only finite time.

hzhou321 10y ago
Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you? Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?
- zeeboo 10y ago
  If I start at 0 and successively add 1, do you agree that I eventually hit any positive integer you could pick after a finite number of steps? Does that not prove to you that I hit every positive integers? Which one do I not hit?
  - hzhou321 10y ago
    You will only eventually hit any given integer after finite steps. It does not prove you will hit every positive integer. You'll miss those that one never can finish giving you -- for example, I'll start with digit 1, and I'll infinitely adding 1 behind it (never finishing). It is an infinite natural number that you can't hit within any finite number of steps.
    - cvoss 10y ago
      The game itself is certainly not the proof that is required. The proof in question is a proof that the game cannot be won by the adversary. By no means do you need to play all possible rounds of the game to conclude this.
      - hzhou321 10y ago
        The discussion really helps me understand the problem. Real numbers are infinity in disguise. Because infinity is not really defined, real numbers are not really defined. Just saying infinity is not finite does not make much sense (in the sense of adding anything helpful). Any real number that you can finitely describe can be included in a finitely described counting scheme. Let's use the counting schemes of rational numbers, I'll hit arbitrary numbers to arbitrary precisions. Taking a limit, it is not clear that I miss any numbers (including pi). To defeat this scheme, the adversary has to keep adding digits to his real number, as well as keep shrinking his allowed precision infinitely. Now both the number (that is being infinitely being described) and my counting process is infinite, we didn't and can't prove anything. There is simply no conclusions to be drawn.

jonreem 10y ago

A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it. So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function. We can extend this and say that the set of even natural numbers is countably infinite because it has a mapping function of x => x / 2. The same is true for all integers (natural numbers + negative numbers): x => if (x < 0) { x * -1 * 2 } else if (x == 0) { 0 } else { x * 2 + 1 }, i.e. if it's negative map it to an even number and if it is positive map it to an odd number, if it's zero map it to itself. You can even write a function that maps all rational numbers to the natural numbers, since each rational number can be written as a fraction of two integers. (Figuring out the function is a fun exercise but it is also easy to google) However, you can't write a function that maps any real number to a natural number. The easiest to understand proof of this is Cantor's Diagonal Argument[0], which is a proof by contradiction that shows that any attempted function must exclude some real numbers. Therefore, the real numbers are not countably infinite, and we call them uncountably infinite. EDIT: In response to your edit, Cantor's Diagonal Argument basically shows that for any given function (and you have to define the function completely ahead of time - that's key) I can give you a real number that is not included in the domain of your function. [0]: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

hzhou321 10y ago
Why a pre-definable function matters here? We have allowed a real number (that cannot be exhaustively described) in, so why a real function (that you cannot exhaustively define) has to be excluded? Isn't it unfair that I am only allowed to use a finitely definable function while you can choose a non-pre-finitely-describable real number? Isn't that a loop proof -- that the set of real numbers is uncountable simply because it is defined to be uncountable, and it is larger infinity than that of natural numbers simply it is defined to be bigger (as what bigger means in the context of infinity is otherwise undefined).
- zeeboo 10y ago

drdeca 10y ago

> Since you never can finish counting Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer. The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something. Also, the specification can't depend on things like, what reals have been given as input "previously". Say that in this game, you have two obligations: 1: for any n,m I give, you must tell me what the mth binary digit of the real number associated with n is, in the mapping you are considering (or, if there is no real number associated with that natural number.). 2: For any finite collection of the binary digits of the real number I am choosing, you must tell me the first natural number such that the corresponding real number matches all the digits I specified. With these rules, I can choose my real number (specifying more and more digits) such that for any natural number, I will be able to show that my real number doesn't correspond to that natural number, or any lower one. Therefore, there is no natural number that corresponds to my number in the matching system you are providing. (To win, I repeat this procedure: Ask what the first natural not ruled out already by my specification of my number is. (Call that n) Ask what the nth digits of the real associated with n is. Inform you that the nth digit of my number turns out to have the other value for its nth digit, so no natural less than or equal to n corresponds to my number. Repeat. This will only ever tell you more about my number, and will not result in me changing my mind about any of the digits of my number, yet there is no natural number which won't ever be ruled out as potentially corresponding to my number. Therefore, no natural corresponds to my number in your system. So I win.)

hzhou321 10y ago
Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?
- drdeca 10y ago

hardboiled 10y ago

Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell's 'Primer of Infinitesimal Analysis' has good details. The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes. Calculus can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve. Also you don't differentiate numbers (real or rational), you can only differentiate functions. Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.

hzhou321 10y ago
Seems I agree with you (or you agree with mine) :) I don't have problem with calculus -- or I wouldn't be able to do physics. I am having problem with calculus based on infinity. > Also the idea of an actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent. Well said. > Also you don't differentiate numbers (real or rational), you can only differentiate functions. Of course I meant distinction.

esoterica 10y ago

> Since you never can finish counting You seem to have serious trouble wrapping your head around the concept of words having different meanings in a layman context and a technical context. Just because you can't "count" (in the counting-out-loud-on-your-fingers sense) the naturals doesn't mean they aren't countable in the rigorous, set-theoretic sense. Your entire argument basically boils down to "this word means something different from what I'm used to it meaning in daily conversation". > I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. How narcissistic do you have to be to assume that an entire academic discipline must be wrong just because you can't personally grok it? > I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity. It isn't hand-waved at all. Just because you don't understand it doesn't mean it's not rigorous. Go read a set theory textbook or take a MOOC or something.

jacobolus 10y ago
I downvoted and flagged your comment. Please don’t bring gratuitous personal attacks here.

raattgift 10y ago

I'll try to physicalize countability. Start counting the naturals: 1, 2, 3, ... At future timelike infinity you'll reach infinity. Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. 0 0.0........ At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real. This survives across changes of positional counting systems, and almost certainly survives arbitrary choices of non-lossy notation, as long as you start with a finite representation of 0.

stymaar 10y ago
> I'll try to physicalize countability. > Start counting the naturals: 1, 2, 3, ... > At future timelike infinity you'll reach infinity. > Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. > 0 0.0........ > At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. > In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real. The same reasoning applies for rationals, yet they can still be counted. The definition of «can be counted» means there is a bijection between your set and the naturals. Such kind of bijection can easily be created for the rationals[1] and Cantor's diagonal argument[2] shows that you can't create such bijection for reals. There is nothing really intuitive about this concept, but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. [1] https://en.wikipedia.org/wiki/Pairing_function#/media/File:Diagonal_argument.svg https://en.wikipedia.org/wiki/Pairing_function#/media/File:D... [2] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- hzhou321 10y ago
  > but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. It is only straightforward once you accepted infinity and all other definitions/description based on infinity. If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical. >> Start counting the naturals: 1, 2, 3, ... >> At future timelike infinity you'll reach infinity. This highlights the flaw. You reach infinity with infinity. Nothing is really being said about infinity. But somehow if you accepted the understanding infinity here, the rest of thesis such as cantor's diagonal argument may seem to be natural, except you forget that you really didn't know what infinity is. All property of infinity cannot be finitely described. So anything about infinity is built on top of infinity. Turtle all the way down (or up), and we don't really know what it is.