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If we want to talk about uncountable sets, then we can distinguish between objects in the set that we can actually "specify" and those that we can't. An example of objects that can be specified are $5$, $\sup_xf(x)$, and Chaitin's constant. I can't give any examples of objects that can't be specified, since I would have specified them in doing so.

This notion is formalized in some specific contexts: E.g. a definable real number, or a definable set (of natural numbers). Based on a few quick google searches I could only find these two examples. I'd just like to get a pointer to some literature that I can read about it.

Is there a general notion of "definability" of arbitrary mathematical objects?

  • In particular, in classical mathematics. In constructive mathematics the notion of definability collapses to computability afaik.

  • I'm particularly interested if it's in type-theoretic foundations.

  • I can imagine there are some difficulties related to logical paradoxes when we try to define a notion of "definability" in full generality?

user56834
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    Definability is always relative to some context – what formal system you work with, what axioms you assume, what you allow as parameters, etc. – Zhen Lin Mar 20 '21 at 14:34

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For any structure $\mathfrak{M}$ there is a general notion of (first-order) definable subset/function/relation/element of $\mathfrak{M}$, provided by the (first-order) formulas in the language of $\mathfrak{M}$. We can also talk about definability with parameters, and so on. See here for a quick summary of this.

Now note the "first-order" bit above. This refers to the specific choice of first-order logic as our framework for making and evaluating definitions. There are other logics out there - second-order logic, infinitary logic(s), etc. - and each logic gives rise to its own definability notion. If you're interested, I strongly recommend the (legally freely available online!) wonderful collection Model-theoretic logics.

As to meta-issues, for any "reasonable" logic $\mathcal{L}$ there is a general obstacle (usually phrased for first-order logic, but much more generally applicable than that) to talking about the $\mathcal{L}$-definable phenomena in a structure $\mathfrak{M}$ within that structure itself. The existence of pointwise-definable models of set theory provides a particularly powerful example of how strong this limitation truly is! However, this doesn't prevent us from performing this analysis from outside $\mathcal{M}$. So we just have to be careful what statement is being made where.

Noah Schweber
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  • Is there a difference between what is definable in two sufficiently general logics? I would guess that there is an analogous notion of "Turing completeness" for definability? i.e. any reasonable candidate language for foundations of mathematics must be able to encode Set theory, and hence define an object by defining it in terms of this set theory encoding. Such a language would be "Set theory complete" and hence be ably to define anything that can be defined in set theory. So there is an "objective" notion of "definability" similar to "computability"? – user56834 Mar 20 '21 at 14:45
  • @user56834 no, different logics have wildly different "levels of definability," and there is no "strongest logic" in any good sense. Re: "any reasonable candidate language for foundations of mathematics must be able to encode Set theory," keep in mind that the standard set-theoretic framework $\mathsf{ZFC}$ is incomplete, even with respect to first-order sentences, per Godel; being "rich enough for math" doesn't imply maximality in any sense. – Noah Schweber Mar 20 '21 at 14:54
  • (Incidentally, when I say "logic" here I'm very deliberately being a bit vague about what I mean. Since it's such a loaded notion, there are many different competing formal notions of "logic"/"formal system"/etc. If you want to pick one to stick with, my standard choice is the notion of regular logic - see the first chapter of the collection linked in my answer for the definition, or the last section of Ebbinghaus/Flum/Thomas.) – Noah Schweber Mar 20 '21 at 14:55
  • That said, there is indeed a sense in which first-order logic is "special:" it is a maximally strong logic with respect to two particular "tameness" properties, namely compactness and downward Lowenheim-Skolem (the former can also be replaced with having recursively enumerable validities). This is Lindstrom's theorem. – Noah Schweber Mar 20 '21 at 14:59
  • So where does my intuition go astray? My thought is: The notion of "definable" should capture: anything that a human (or smarter being) could possibly refer to using some kind of formally defined expression. Surely there must be some kind of language that is "complete" in the sense that, any expression that we could write down could be interpreted as an expression in that language. If not, but humans still have a "mental"/"practical" language in which the expression is well-defined, we can simply extend the language to incorporate that. – user56834 Mar 20 '21 at 15:04
  • @user56834 Well for starters I disagree with the very first premise, that the mathematical concept of definability should somehow be limited to what humans can do even in principle. There's no claim, for example, that second-order logic is at all "human-capturable" in any particularly good sense. Even granting that, though, I don't see how the rest is justified: the various semantic paradoxes should convince you that the language we use naturally to talk about mathematics is extremely fraught. (Really, the stability of the notion of computability should be surprising, in my opinion.) – Noah Schweber Mar 20 '21 at 15:06
  • Let me ask for a clarification: Is it correct to say that it follows that there is no single "strongest notion of definability", from the Tarski's undefinability theorem? i.e. does the theorem imply that for any (computable) language (i.e. a language for which we can actually decide with a TM whether an expression is syntactically correct), there is another language within which additional objects can be defined? – user56834 Mar 20 '21 at 15:32
  • @user56834 You don't even need Tarski for that if you're restricting attention to "computably presented" logics. Simply look at a structure with uncountably many automorphically-fixed subsets and use a counting argument (one of the standard requirements for a logic is that its satisfaction relation be isomorphism-invariant, so it's not enough to have uncountably many subsets - only automorphically-fixed subsets are definable by some logic, in the first place). For example, $(\mathbb{N};+,\times)$. – Noah Schweber Mar 20 '21 at 15:36
  • @user56834 Coming back to this much later, did my previous comment address your clarification question? – Noah Schweber Aug 18 '21 at 05:28
  • Honestly, I didn't understand it. I was thinking about it for a while and then ended up dropping it and forgetting to respond. It seems to me that maybe the term "definability" is used differently from what I have in mind, but I don't know enough to verify this. I might read up on this at some later point. – user56834 Aug 18 '21 at 07:25