Proof checking algorithms

Question

Is it possible to develop a finite algorithm that can check the validity of any mathematical proof? I'm trying to relate in the sense of halting problem Note- I am a post graduate math student who has pretty much no sophisticated knowledge of theories relating to halting problem or computability

Answer 1

Yes, this is possible.

Simply put, a proof is a finite sequence of statements (each again of finite length) $\phi_1,\phi_2,\ldots,\phi_n$, where each $\phi_k$ follows from some preceding statement(s) by one of finitely many inference rules, or is one of finitely many axioms. Each of these options can be checked systematically.

Admittedly, this was an oversimplification:

• Inference by hypothetical derivations (assume $A$, based n this assumption, deduce $B$; in summary this is a proof of $A\to B$) deviates a bit from the simple procedure I described above. But that can be handled by "carrying" all current hypotheses with each statement and respecting these in the checks
• In most theories, there are infinitely many axioms. But they are always organized in finitely many axiom schemes, which can be checked systematically in finite time. (Indeen, if we could not even decide if a statement we see is an axiom or not, that'd be a very useless theory)
• All this does not apply to any proof you may ever read in any book or paper. Instead it applies only to proofs broken down to the moste elementary steps thinkable ad nauseam. E.g., it would take many pages to completely formally prove that $2+2=4$.