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I want to prove in the following way there is a statement that a computer can't prove. Suppose the time of life of our universe is limited by $t$ and the speed of calculations is limited by $s$. Now, consider the axioms of Peano arithmetic (PA). Can we show that for a given $n$ (in particular, $n=ts$), there is a sentence $S(n)$ provable in PA, such that the minimal number of strings in a (Hilbert-style) proof of $S(n)$ is greater than $n$?

Bertrand Haskell
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    Ask the computer to find a nontrivial integer solution to $x^n+y^n=z^n$ for all $n\ge 3$. There are infinitely many $n\ge 3$. – Dietrich Burde May 22 '23 at 18:48
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    Godel's incompleteness theorem gives you a stronger result: if you have a computer program that only outputs true statements about natural numbers, then there is a true statement about natural numbers that your program will never output. And if you have one unprovable statement, you can get infinitely many more by taking its conjunction with other provable statements. – Karl May 22 '23 at 18:56
  • Also, what do you mean by "shortest form"? When are two forms considered equivalent? – Karl May 22 '23 at 18:59
  • @Karl I've tried to consider FOL-sentences, then we mean they are equivalent (in terms of FOL including all possible interpretations)... – Bertrand Haskell May 22 '23 at 19:02
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    Hmm, would you consider $s$ to be proved if $s$ and $t$ are equivalent and you have a proof of $t$ but not a proof that $s$ and $t$ are equivalent? – Karl May 22 '23 at 19:03
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    Have a look at the proof sketch for Gödel's first incompleteness theorem. In this Wikipedia article, the formula $P(G(P))$ is an explicit example for exactly the kind of unprovable statement you are looking for. This construction works for every system of axioms that can describe the Peano arithmetic. – Andreas Tsevas May 22 '23 at 19:08
  • @Andreas thanks for this link (I really need to study it), but I mean more the existence of sentences which can be proven but their minimal length is sufficiently large... Also (but this is another question), I am interested in "lower bounds" of the length of a proof. – Bertrand Haskell May 22 '23 at 19:15
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    See e.g. how long can proofs be? – Karl May 22 '23 at 19:17
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    If you're just looking for very long statements, try things like "$1\ldots 1 + 1\ldots 1 = 2\ldots 2$". – Robert Israel May 22 '23 at 19:19
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    Once again, Gödel to the rescue, he did a lot to answer these kinds of questions :D See here: "Theorem 5: Given any recursive function f there are provable sentences φ of arithmetic such that the shortest proof is greater than f(⌈φ⌉) in length". And, in general, see the description of Gödel's speed-up theorem on Wikipedia. – Andreas Tsevas May 22 '23 at 19:20
  • @Karl I've totally edited the question. Now, it's indeed more about the proof's length :-) – Bertrand Haskell May 22 '23 at 20:03
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    This may appear to be an off-topic comment (but it bears on the mathematical facts that are relevant to you, so I think it is really on-topic). You want to show that "there is a statement that a computer can't prove" based on physical limitations for computation in a universe that is postulated to be finite. Are you requiring the statement to be physically realisable in some way but that the proof is not physically? If so, what you are looking for sounds like a delicate balancing act. – Rob Arthan May 22 '23 at 21:03
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    There are infinitely many provable statements, only finitely many proofs with length at most $n$, and each proof proves at most one statement, so there is definitely a statement that has a proof but all its proves have length more than $n$. – mihaild May 22 '23 at 21:35
  • @RobArthan thanks for your comment! I agree there is a kind of uncertainty (which may be due to the relativity of a human's knowledge). I just try to define the computational limitations, understanding this definition is fully conditional. In some sense, I even say that, in spite a computer can prove a sentence, we do not accept this proof untill it is not short enough... – Bertrand Haskell May 22 '23 at 21:41
  • @mihaild your argument is really elegant! – Bertrand Haskell May 22 '23 at 21:47
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    You can't prove a statement without writing it out, so mihaild's argument follows from the even simpler fact that there are arbitrarily long provable statements, such as those suggested by Robert Israel ($1+\dots+1=\dots$). If you want to exclude these boring cases, you need to look for long proofs of short statements, as Rob Arthan suggests. This is where the proof length function in how long can proofs be comes in. I think that post answers your question. – Karl May 22 '23 at 21:54
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    You actually could prove statements without writing them down if you used a calculus of proof terms with proposition inference analogous to Hindley-Milner. Then, there are proof terms whose 'principal proposition' has size $O(2^{2^n})$, where $n$ is the size of the proof term. The usual examples are also not interesting, though. They look like $P → P ∧ P$, $P → (P ∧ P) ∧ (P ∧ P)$ and so on (at each step, replace each $P$ in the consequent with the entire consequent). – Dan Doel May 23 '23 at 16:11
  • @DanDoel I think the similar idea is as follows. Consider the generative grammar with two inference rules $S\rightarrow aS$ and $S\rightarrow e$. We can easily prove by induction that this grammar produces any word $a^n$. Now, interprete such words as theorems and sequences of rules (like 1112) as proofs. Then there is a theorem with its proof which has a prescribed (minimal) length, but we know about this fact implicitly, reasoning in an outer formal system by induction. – Bertrand Haskell May 23 '23 at 18:09

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