Slightly confused by this. I wish I knew what more to add, but I know it's asking something about the halting problem, and something about Godel but I cannot seem to figure it out.
2026-04-09 00:00:40.1775692840
Can we program a computer to check a proof in first-order arithmetic is valid, and can we find out whether such a statement is true?
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It is quite possible to program a computer to recognize valid first-order proofs presented to it, if we can accept that the proofs have to be written down in a computer-friendly notation.
This is not directly connected to whether one can recognize true statements of first order arithmetic.
A machine that recognized arithmetic truth would be able to solve the halting problem, since (thanks mainly to Gödel) we know how to formulate the claim "Turing machine number such-and-such halts" as a first-order arithmetic sentence. Since we know the halting problem is undecidable, there cannot be an algorithm that recognizes first-order arithmetic truths either.
(Alternatively, you could note that an algorithm that recognizes truth can be viewed as a powerful proof system for first-order arithmetic, and if it recognizes all truths it would be a complete proof system, which Gödel's incompleteness theorem tells us is not possible. So the first-order arithmetic truths cannot even be recursively enumerable!)