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My question stems from this post - Informal proof of Gödel's second incompleteness theorem

Basically the proof of second incompleteness theorem goes like this:

Say G is the Godel statement of T. Cons(T) => G

So if we manage to prove Cons(T) within T, then we have a contradiction and hence it is inconsistent.

But then we also know that if a system is inconsistent then every statement is provable. Basically,

$\neg Cons(T) => G$

So if Cons(T) => G and $\neg Cons(T) => G$

then G should be true independent of whether Cons(T) is true. Why cannot we formalize that argument within T to prove that G is true without asserting Cons(T)? What am I missing?

Prunthaban
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  • You have to distinguish between what a theory proves and what a theory proves it proves - these don't actually coincide! This is a very good (and common) question, but it is essentially the same issue as in this question (which phrases it in terms of specifically $T=\mathsf{PA}$). – Noah Schweber Jul 21 '21 at 02:03

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