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Noson Yanofsky is a theoretical computer scientist at Brooklyn College. He presents the following argument on pages 329-330 of his book The Outer Limits of Reason, published by the MIT Press.

  1. The set $\mathbb{N}$ of natural numbers has uncountably many subsets.

  2. Let $x$ be a natural number and $S$ a subset of $\mathbb{N}$. Exactly one of the following statements expresses a mathematical fact: (a) $x \in S$, (b) $x \notin S$.

  3. It follows from (1) and (2) that there are uncountably many mathematical facts.

  4. Let $T$ be a first-order theory. Then $T$ has only countably many formulas.

  5. It follows from (3) and (4) that there are more mathematical facts than formulas in $T$.

  6. Therefore $T$ cannot express, much less prove, all mathematical facts.

For Yanofsky's own words, see the article: "Most truths cannot be expressed in language."

I have heard people say similar things on podcasts. They try to explain Gödel's first incompleteness theorem as stating that there are more truths than proofs.

Question.$\ \ $Is Yanofsky's argument valid? If not, why?

It seems to contradict the answers to the following questions: 1, 2, 3. However, Yanofsky seems to be an expert in this area, and his argument was published by the MIT Press.

Note that Yanofsky writes:

"This is all about mathematical facts – not about what can be stated. A mathematical statement is a mathematical fact that can be put into symbols. We saw above that... there are countably [many] mathematical statements. Hence there are massively more mathematical facts than mathematical statements."

Perhaps this could be made more precise by encoding mathematical facts as sets.

Asaf Karagila
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simple jack
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    Gödel's incompleteness theorem is not a counting argument. It proves that one countable set which apriori is a subset of another countable set is a proper subset. The properness is not proved by showing they have different cardinality. – Chad K Jun 26 '23 at 08:41
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    what Yanofsky says is true but not remarkable. Godel's incompleteness theorem makes a far more interesting and informative assertion about actual statements that are true but not provable. see https://math.stackexchange.com/questions/4722030/is-there-a-language-that-definitely-wont-have-inconsistencies/4722065#4722065 – RyRy the Fly Guy Jun 26 '23 at 15:54
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    It could be expressed with reals. There are uncountably many reals, even just in an interval like $[0, 1]$. About any given real number $r$, there exist facts, ... and so it goes. – Kaz Jun 26 '23 at 17:59
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    Yanofsky's argument is valid and, to be fair to him, he does not use it as an explanation of Gödel's results. However, it seems rather banal in context. The work of Chaitin and Calude that he refers to seems to be far more interesting, but he doesn't enlarge on that. – Rob Arthan Jun 26 '23 at 22:21
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    Seeing as a mathematical truth may have infinitely many proofs or none at all, the significance of counting proofs is not clear. – bof Jun 26 '23 at 23:14
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    This seems to be a simple restatement of the fact that we can't write down or define most subsets of the natural numbers. Similarly we can't explicitly write down most real numbers. Both cases are a simple counting argument of uncountably versus countably many. – quarague Jun 27 '23 at 09:14
  • @JohnDoe: Yes, but more generally, Lawvere's fixed-point theorem certainly can be used as a counting argument, e.g. when instantiated as Cantor's theorem. – Corbin Jun 27 '23 at 12:45
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    To quote Joel David Hamkins on MO: "The concept of definable real number, although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers... ." That answer is about real numbers but it applies equally well to this question if you replace "real number" by "subset of the naturals". I recommend giving it a read. – Carmeister Jun 27 '23 at 15:52
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    There will always be just a finite number of mathematical proofs, each of them constructed by someone. The set of all non-constructed proofs is empty. – Lehs Jun 27 '23 at 23:03
  • We could also say that there are continuum many facts of the form $x>y$ (where $x,y\in$ ℝ), while only countably many of them can be expressed as statements in a theory of ℝ with a countable language. It's not quite clear what definition of “_mathematical facts_” is used by Yanofsky. Perhaps, in my example, they are just ordered pairs, the elements of the $>$ relation over ℝ. Or, perhaps, they are the true statements in a formal theory (with an uncountable language) that has a constant symbol for every element of ℝ. Anyway, this observation seems rather trivial and not very consequential. – Vladimir Reshetnikov Jun 30 '23 at 21:12

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That's correct, but it is not very interesting: we don't have "access" to most mathematical facts. It is not even clear what a proof of an arbitrary mathematical fact would mean, because to ask for a proof of something we have to write it down, and there are only countably many strings we can write.

Gödel's theorem is much more interesting: it says that there is a formula that is independent of PA. So you can write some statement in arithmetic language (and it actually can be as simple as "this Diophantine equation has a solution") and neither it nor its negation can be proved in PA.

Compare it with, say Euclidean geometry. There are uncountably many lines and points, so there are an uncountable number of "facts" like "this point is / is not on this line". But Euclidean geometry is complete. So Gödel's theorem shows a difference between arithmetic and geometry.

And also there is (assuming consistency of ZF) a model of ZF where there are, from an outside point of view, only countably many sets.

mihaild
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    Furthermore, if ZF is consistent, there are pointwise-definable models of ZF (models of ZF in which each object is the unique object satisfying some formula). – Gavin Dooley Jun 26 '23 at 21:27
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The argument is fallacious.

We can use the same basic argument in an even simpler setup, which clearly indicates what goes wrong.

There are uncountably many subsets of the natural numbers. There are only countably many ways we can express a statement of the form $\pi\not\in S$ for some $S\subseteq \mathbb N$. So there are massively more true facts of the form $\pi\not\in S$ than we can even express, let alone prove.

But of course, we can prove all of them, since we can prove $\pi\not\in \mathbb N$ and hence $\pi\not\in S$ for every $S\subseteq \mathbb N$. The fallacy is that a single proof can prove uncountably many facts, so being able to state a fact is not a prerequisite to proving it.

The quote "most truths cannot be expressed in language" is correct, at least for this definition of "truth". But it doesn't follow that they can't be proved.

Especially Lime
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    We can prove that $\forall S \subset \mathbb N: \pi \not \in S$, but such proof isn't proof of "$\pi$ isn't in set of odd natural numbers": to prove the second from the first, we need to add two extra lines - that set of odd natural numbers is subset of $\mathbb N$ and then use universal elimination. However, for most subsets we can't do this, because we can't define them in the first place to write statement that they are subsets of $\mathbb N$. – mihaild Jun 26 '23 at 23:53
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    "There are only countably many ways we can express a statement of the form π∉S for some S⊆N" Why? Do you mean there are only countable many ways in first-order logic? "The fallacy is that a single proof can prove uncountably many facts" Not in the formal logic sense of "proof". A proof is sequence of statements in which each statement is either an axiom or a statement derived from previous statements through the rules of logic, and the final statement is the statement being proved. So "proof(for all x in X, x has property P)" $\neq$ "all x in X, proof(x has property P)". – Acccumulation Jun 28 '23 at 00:12
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    @Acccumulation Despite not writing a proof for all x in X, you have an algorithm for generating one from the proof of for all x in X, which is just universal elimination. – Passer By Jun 28 '23 at 03:24
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    @PasserBy only if you accept some strong form of constructivism, that every x in X can be defined by formula. If you can't define x you can't use universal elimination. – mihaild Jun 28 '23 at 08:31
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A slightly different perspective - Yes, there are uncountable (trivial) mathematical facts, but you don't need to individually prove each of those facts. What's much more interesting is the fact that you can, in most situations where it matters, formulate statements in the language of first order so that it expresses, in theory, uncountable number of facts.

For example, proving that some function function $f: \mathbb{R} \to \mathbb{R}$ is continous. Which is equivalent to the statement that for each real $x$ (clearly uncountable number of $x$), $f$ is continous at $x$. We can obviously prove statements such as this, and so this shows that not only can we, in effect, state uncountable number of "facts," but also prove such "facts" as "facts" in a finite number of steps.

In a nutshell, though, Yanofsky's argument is just a variation of zeno's paradox.

Bertrand Wittgenstein's Ghost
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    I think most people would be more comfortable with the phrasing "for each real $x$, $f$ is continuous at $x$" rather than "for each real $x$, $f(x)$ is continuous" – Stef Jun 27 '23 at 07:52
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    It's more useful to view Yanofsky's argument as a flavor of Gödel's First Incompleteness, which in turn can be viewed as a specialization of Lawvere's fixed-point theorem. See e.g. Yanofsky 2003 for a holistic treatment. – Corbin Jun 27 '23 at 12:48
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    @Corbin But Yanofsky's argument has nothing to do with Godel's incompleteness theorem. It's not "a flavor" of it, it's just unrelated. – Noah Schweber Jun 27 '23 at 15:14
  • @stef You are correct. That was a silly oversight from me haha. – Bertrand Wittgenstein's Ghost Jun 27 '23 at 15:50
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    @NoahSchweber: This question is about whether there are more true sentences than provable sentences. Gödel constructed an unprovable-yet-true sentence. They're more closely related than one might think. Either way, I think we agree that it's not an instance of Zeno's paradox. – Corbin Jun 27 '23 at 15:55
  • @Corbin 1: There are uncountable number of "facts." 2: Any FO theory T has only countably many formulas. 3: T cannot possibly express all facts. Zeno: Motion across some distance D implies that one must constantly halve D, but in order to travel a distance of D, we need to halve D, and each subsequent result (which is infinite); therefore, it's impossible to keep halving D until we get to O. Hence, there can be no motion. – Bertrand Wittgenstein's Ghost Jun 27 '23 at 16:10
  • @Corbin Zeno's issue: you don't need to halve D and each subsequent result to get motion. Yanofsky's issue: I'll let you take this one over. – Bertrand Wittgenstein's Ghost Jun 27 '23 at 16:11
  • That said, I might be off in my analysis of the issue - But that's the reasoning behind my conclusion. – Bertrand Wittgenstein's Ghost Jun 27 '23 at 16:19
  • @BertrandWittgenstein'sGhost: I see your reasoning. I think that the topological circumstances are different; you're talking about a continuous setup, but Lawvere (and thus Yanofsky) are using pointful category theory, and Lawvere was generalizing folks like Cantor, who were definitely talking about sets of natural numbers. – Corbin Jun 29 '23 at 14:26
  • @corbin, I understand where you are coming from, the topological circumstances are most definitely different, but I don't think that takes anything away from my argument. The issue I take with Yanofsky is that his argument is structurally fallacious (Definitely not invalid). Regardless of how one was to flesh out the details of some argument, if the underlying structure is fallacious, then the argument is necessarily fallacious. This is all there is to what I am saying. I do appreciate your point of view, though. – Bertrand Wittgenstein's Ghost Jun 30 '23 at 01:36
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Nobody seems to have mentioned the following point. Let $\mathcal F$ be the principal ultrafilter on $\mathbb N$ generated by $1\in\mathbb N$. Then the assertion $(\forall A\in \mathcal F) (1\in A)$ includes uncountably many "truths". That's all Yanofsky's argument is saying. It is difficult to derive meaningful insights from such a platitude.

Since Zeno's paradox is far from a platitude, it is clear that Yanofsky's argument has nothing to do with Zeno's paradox.

Mikhail Katz
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I slightly disagree with the assertion that there are uncountable many facts, as only a countable many of these facts can actually be stated. That is, we can only describe a countably amount of subsets of $\mathbb{N}$.

Per Alexandersson
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I would say (and I'm aware of where I'm saying this) that there are no proofs.

There are only facts, and statements about facts which are true or false or perhaps undecidable; And for each of the (uncountably many) statements about facts it can be shown that they are true or false or undecidable, but that adds nothing1: They have always been true or false or perhaps undecidable.

The opposition of "statement" and "proof" is a fallacy. Showing that a statement is true or false or undecidable doesn't change anything about the statement. There is always a 1:1 relation between the statement and its quality of being true or false or undecidable.

1Adds nothing to the statement, or the facts. Of course it gives us certainty and with it the tools to discover more facts, and is very valuable in this respect.

Peter - Reinstate Monica
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