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Since the entire observable part of the universe can only be in a finite number of physically distinguishable states, it seems rather strange that an efficient formal description of the universe would require the notion of uncountable sets.

Just like we're better off accepting that infinitesimal numbers do not exist, rather than developing an unwieldy formalism in the form of non-standard analysis, one can consider if we would be better off doing away with uncountable objects.

From what I have read, people have argued in similar ways, but not much progress has been made. So, what is holding up the development of a better analysis based on only computable quantities?

Count Iblis
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    There are a rational and computable calculus IIRC. However modern mathematics isn't concerned with modeling the universe or having much connection to it at all so I think it's not really a legitimate question. – Cameron Williams Apr 30 '15 at 19:09
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    Whence came your assertion that "the entire observable part of the universe can only be in a finite number of physically distinguishable states"? – user225318 Apr 30 '15 at 19:16
  • @CameronWilliams While math is obviously not limited to its application in theoretical physics, it is still the case that in theoretical physics we use the standard mathematical formalisms, also a lot of modern math is inspired by developments in theoretical physics. – Count Iblis Apr 30 '15 at 19:22
  • @user225318 this follows from quantum mechanics. E.g. Church–Turing–Deutsch principle: "The principle states that a universal computing device can simulate every physical process." – Count Iblis Apr 30 '15 at 19:26
  • You need to be very careful about the chicken and the egg. Most of the principles of quantum mechanics (operators in Hilbert spaces) are developed under the assumption of the standard sets of real numbers. If you are only allowed to work with the ultrafinitist's version of the real numbers, "Hermitian operators" are no longer guaranteed to be diagonalizable, thus you lose the notion of pure states, and it is entirely unclear to me how this effects quantum computation. – user225318 Apr 30 '15 at 19:34
  • Furthermore, the reason that the CTD principle relies on quantum universal computational devices is precisely the acceptance that classical, finitary Turing machines are incapable of simulating all physical processes. So what you want is to take the CTD principle to be a given "truth" when the usual finitary Turing machines are considered. You can take that as your philosophy, but I don't see it as a given, or a justification that "the entire observable part ... finite number of physically distinguishable states." – user225318 Apr 30 '15 at 19:38
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    A person, who is sufficiently smart, would find the universe in a single state I believe. Should we do away with all integers too? – ClassicStyle May 03 '15 at 04:23
  • There are probably other questions discussing the use of the real numbers in analysis, similar to the question here. I picked the first one that came to my head. – Asaf Karagila May 03 '15 at 04:35

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I can see what you mean by doing away with the uncountable reals, just to discard all the superfluous ones that make the set uncountable rather than countable, but there are a couple things that should be noted. Also, you seem to imply this can be accomplished by using only the computable reals. That is known to not work, which I discuss later, so I will focus on doing away with the uncountable reals, as you say.

First, the reals are uncountable as defined in standard set theory relative to that theory. Countability is a relative thing, the existence of a bijection depends on what functions are available. So whether or not a bijection exists between the reals and the integers depends on what bijections the theory makes available. As a matter of fact, it is generally accepted there is a countable model of ZFC, which would define the set of reals to be countable (relative to some meta-context, not relative to the ZFC in which the set is defined). However, that does not help much, because reals defined as part of such a model still leave real numbers that cannot be identified in any meaningful way.

Second, it is not all that clear what reals make the set of reals uncountable. What countable subset do you accept, while rejecting the others? This is tricky. For example, if you decide to only accept real numbers defineable in some language, you can use a straightforward definition that goes through definable numbers in that language to unambiguously define a real number not defineable in the language.

Alternatively, If you require each digit to be computable, you loose the least upper bound property, which is a pretty big deal.

So, you see, there is no obvious way to reject the superfluous reals. To be clear, it is not known to be impossible to do so without major problems, it is just not a simple task.

So, you ask what is holding up progress in defining reals in a way that makes them countable and retain all their expected properties? I'd say, first off, that there are not enough people asking this question! Additionally, it is a really hard problem. You can be sure that thousands of hours of brilliant thought have gone into it, especially back when uncountability was viewed with suspicion, in the early days of the idea.

I absolutely agree with you, I think the reals would more naturally describe reality if they were defined so as to be inherently countable, but at the same time retain all their properties. And it is absolutely not known that this is impossible. It is just that most mathematicians are comfortable with the reals as they stand, and not a lot of work is being done on making this happen.

jack
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Short answer: no.

When we define the real numbers we do so in such a way that we define nice properties so we know what we are talking about. We do so in such a way that all the numbers we construct will be related to the language and logic we use.

Whether a number is constructible or not depends on the features of the language and logic we choose to deploy. Whether a number has ever been constructed or named is a contingency of history. Whether a particular number will ever be constructed is, in general, unknown.

We have lots of numbers we will never need in order to say useful things about the numbers we will need. We don't yet know which these will be .

Mark Bennet
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  • It is inaccurate to say that "whether a number has ever been constructed or named is a contingency of history". That would imply that all reals could be constructed or named, which would imply that the reals are countable. Also, you did not justify your answer of "no", you just said why the reals are currently defined in a way that makes them uncountable, not why this has to be the case. – jack May 03 '15 at 05:58
  • @jack That wasn't quite the point I was making. In fact only a finite set of numbers will ever be constructed, even though there are more which are potentially constructible. We do not know which these will be. The particular numbers which are constructible depend on the rules you allow - has $\pi$ been constructed? Or has it just been named? To describe a number which cannot be constructed is to give the information which is necessary to name it. The numbers which have already been named have depended on what people have found interesting: a contingency of history. – Mark Bennet May 03 '15 at 06:08
  • Yes, but even numbers which are identifiable in the most liberal sense form a countable subfield of the reals. Think of all images of such descriptions written down. All images of any size in a particular format are countable. So yes, the subset of these that is identified is a contingency of history. These numbers, however, are not OP's concern as I understand it. It is all those numbers that cannot, even in principal, be identified in any meaningful way. And these are all real numbers other than an infinitesimally small portion of them. – jack May 03 '15 at 06:28
  • @jack Indeed - I have used naming rather than construction in my last comment - my point is that we have not even named all the elements of $\mathbb N$ let alone $\mathbb Q$, so anything like a subfield of the reals is already an abstraction beyond what has actually been constructed or ever will be. The idea of a field gives us a context in which to talk about the numbers we have constructed. So does the concept of the reals. We use the abstraction because we find it useful. – Mark Bennet May 03 '15 at 06:47
  • I think the problem is that there are reals that can't be named in principal, regardless of time and resources. This is not the case for the naturals or rationals. If a number can be named but has not, that is a matter of experience. If a number cannot be named, that seems more related to the nature of the number. In fact, such a number seems so divorced from reality that it tempts the question of whether a system that permits such a number is optimal for describing reality. I think this is OP's position, more or less. – jack May 03 '15 at 07:10
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I don't know what "uncountable" means; I can count any specific real, like $\pi$ -- there, I did it, there is one of them.

There are models of $\mathbb{R}$ where all sets are measurable. To do this you must reject the Axiom of Choice, perhaps replacing it with a weaker version.

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vadim123
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  • It is not true that you can "count any specific real". If you could, then the collection of, say, all pdf files that unambiguously identify a real would be countable and would contain the reals. – jack May 03 '15 at 05:54
  • No, it means there are specific reals you cannot identify. That is one implication of an uncountable set. It has members you cannot ever identify in any way. So no, there are specific reals I can't name (because no one can), and so you can't count them. – jack May 03 '15 at 17:36
  • That is a clear misunderstanding of the reals. The reals are defined in a way that there are elements that cannot be defined. If you do not understand this first principal, please refrain from giving answers implying you have an understanding of the topic. – jack May 03 '15 at 18:24
  • I do not mean it as insulting. I feel it is important to only answer questions when one has a reasonable understanding of the subject. If you really do not know that the reals are defined in a way that includes undefineable numbers, you really lack a basic understanding of the subject. That is not an insult, just an observation. And when you answer a question with such an understanding, it misleads users of this site. I just think your contributions would be more appropriate in areas where you have more experience. Or you could study the terms involved. Just please don't throw out answers. – jack May 03 '15 at 18:29
  • You do not own the site, you cannot say who is welcome. And it should be clear that answers from people who do not understand the material are not likely to be correct, and so are inconsistent with the objectives of this site. I don't want you to leave the site, just to do sufficient research and/or focus on questions you have the background to answer. And you don't need to delete your comments, it makes me look schizophrenic. – jack May 03 '15 at 18:33