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What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of mathematics?

Conifold
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Dan Christensen
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  • For anyone who needs a few hints to get the juices flowing (like myself!), the Wikipedia page on definable numbers has quite a few classes of cases where undefinable numbers matter, in case any of them qualify as "important" enough to make an answer from: https://en.wikipedia.org/wiki/Definable_real_number – Cort Ammon Nov 20 '15 at 06:55
  • Do you have an example of such a result? Not sure what the philosophy issue is here, lots of math results make use of unusual things and are perfectly reasonable results. – James Kingsbery Nov 20 '15 at 13:23
  • @CortAmmon Yes, I saw that. No references to anything I think of as real analysis, important or otherwise. Thanks anyway. – Dan Christensen Nov 20 '15 at 18:39
  • @JamesKingsbery No, I don't have a single example. I would post the question at Math SE, but, from past experience, I know they would go bananas, down-voting my question like crazy. I thought that readers here might be less invested in the question. – Dan Christensen Nov 20 '15 at 18:45
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    @DanChristensen PhilosophySE does seem to be more gentle in that regard. However, we may need more information as to what you are referring to. The fact that the Wikipedia page on the topic doesn't include anything you consider to be "real analysis" suggests that the term may be ambiguous enough to warrant an explanation of the meaning you are seeking. (Welcome to Philosophy SE. On Math SE, we down vote you like crazy. Here, we question the meaning of your every word! You just can't win, can you? =) ) – Cort Ammon Nov 20 '15 at 20:55
  • @CortAmmon What is real analysis? Think calculus. See "Real Analysis" at https://en.wikipedia.org/wiki/Real_analysis It is a sub-field of mathematical analysis. See https://en.wikipedia.org/wiki/Mathematical_analysis – Dan Christensen Nov 20 '15 at 21:16
  • Isn't the completeness of the real numbers the simplest and most obvious example? There are only countably many definable numbers, and the continuum must be uncountable. If all we have is definable numbers, the real line has holes in it, as the constructive real line does. – user4894 Apr 12 '16 at 01:54
  • possible duplicate, at least related, [Undefinable Real Numbers](https://math.stackexchange.com/questions/961558/undefinable-real-numbers) – Ronnie Royston Sep 11 '17 at 18:00

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The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in the theory of formal systems because they show how expressive they are, for example Tarski's theorem on the undefinability of truth states that in a consistent formal system that includes basic arithmetic the truth predicate is undefinable (this is closely related to Gödel incompleteness).

Undefinable real numbers are the numbers that can not be uniquely described by definable predicates, so the notion is useful in studying the language of analysis rather than its objects, which is the main subject of the classical analysis. Still, undefinable numbers can help express certain facts about the higher reaches of the universe of sets. My favorite is the undefinable real number 0# discovered by Silver in 1966. Its existence is unprovable in the standard set theory (ZFC), but has a decisive effect on the structure of the universe of sets. If 0# does not exist then the universe of sets looks a lot like Gödel's constructible universe, which is well-behaved and nicely describable, if it does then not only is it non-constructible but generally wild, in particular every uncountable cardinal of the universe is "ineffable" within its constructible part.

Conifold
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Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "important" is that no continuous chaotic system can be perfectly modeled by a Turing machine. Thus, if a truly chaotic dynamic system exists, it could not be part of a simulation of the universe that is running on a Turing machine.

Cort Ammon
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    See Joel David Hamkins's brilliant analysis (his checked answer to the question) of the commonly held opinion that there must be undefinable reals since there are only countably many definable ones. http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb – user4894 Jan 15 '16 at 06:02
  • It is very simple mathematics to conclude from the properties of uncountable and countable that there must be undefinable numbers if the set of reals is uncountable. Hamkins says: "if the ZFC axioms of set theory are consistent , then there are models of ZFC in which every object, including every real number, ..., is uniquely definable without parameters." This proves that ZFC is inconsistent. In fact nothing is uncountable. All diagonals of Cantor-lists are countable. Cantor's argument shows only that his clumsy way of enumerating fails. Other ways do succeed. – Hilbert7 Sep 10 '17 at 20:07
  • @Heinrich: What do you mean by "Cantor's [...] clumsy way of enumerating"? As far as I can read Cantor's diagonal proof, he does not provide any such way, but instead phrases his proof such that it applies to *any* enumeration that could be defined, regardless of how it works. – Kevin Jan 10 '22 at 02:49
  • @Heinrich Did you intend to take a step beyond the typically proven statement that ZFC cannot be proven consistent and indeed make the statement that ZFC is inconsistent? The latter is a much stronger claim, and is not typically accepted by mathematicians. – Cort Ammon Jan 10 '22 at 15:11
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To paraphrase Joel Hamkins answer (pointed out by user4894) on the notion of undefinable reals.

The naive account of undefinability points out there is only a countable number of ways we can describe a number, but there is an uncountable number of reals, hence there must be reals that we can't describe; however the notion of definability is problematical: since the real line can be well ordered (in ZFC) we can ask for the least undefinable positive real; but this defines it, and also by construction it is undefinable and so this has led us straight to a contradiction.

At least part of the problem here is the nature of the logical language we use with ZFC.

The point is that the concept of definability is a second-order concept, that only makes sense from an outside-the-universe perspective. Tarski's theorem on the non-definability of truth shows that there is no first-order definition that allows us a uniform treatment of saying that a particular particular formula is true at a point and only at that point.

Hence the notion of definability is important in suggesting that the first order logic isn't sufficient.

Mozibur Ullah
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I would just like to point out, and I'm doing so in an answer and not in a comment because I do not have sufficient reputation, that the standard "least undefinable number" strategy that @Mozibur Ullah mentions does not work for the real numbers.

In order for this to work, the structure needs to be not just ordered, but well-ordered. JDH (mentioned in Mozibur's answer) talks about ordinals, which are; the real numbers are not.

Because the real numbers are not well-ordered (under the standard ordering, and without the Axiom of Choice, it is possible they have no well-ordering), it is possible that for any real number, there is an undefinable real number less than it, so there is no least undefinable real number.

Jozef Mikušinec
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