Is there any unreachable result?

Question

I hope that this question is reasonable and make sense because I am not sure.

Every theorem's proof is consisting of finite logical steps.

Can a proof of the theorem require infinitely many logical steps ?

If it is so, we can not prove it right ?

Answer 1

There is such a thing as infinitary logics, where a proof can consist of infinitely many steps, suitably arranged. But that is not really mainstream, in the sense that it is not intended to be a useful model of the ordinary mathematical reasoning that is done outside mathematical logic.

More mainstream, you may have heard of Gödel's incompleteness theorem, which says that if you choose some axioms to reason from in a halfway reasonable way (there's a precise technical definition of "halfway reasonable" that applies here, of course), then there will be a claim $G$ such that neither $G$ nor $\neg G$ can be proved. That does not have anything to do with "infinitely long proofs". But the theorem also comes with an argument that $G$ is actually true, which can be said to be an example of an "unreachable result". What actually happens, of course if that $G$ can be proved (in finitely many steps) if only we allow the proof to use stronger assumptions than those of the system the Incompleteness Theorem was proved for.

Answer 2

By the usual definition, a proof is a finite sequence of statements in some language, where each statement is of finite length. However, there is such a thing as Infinitary logic.