Inconsistency and omega-inconsistency (Godel's Incompleteness Theorems)

Question

I am reading Godel's Incompleteness Theorems by Raymond Smullyan. On page 57 of the book, it says that is a system S is simply inconsistent, then every sentence is provable in S, and thus S is omega-inconsistent.

I don't really understand why. Please help me.

Answer 1

Simple consistency and $\omega$-inconsistency are defined at page 56.

The definition says:

if there is a formula $F(x)$ such that the theory proves $\exists x F(x)$ but proves also $\lnot F(0)$, $\lnot F(1)$, and so on, we say that the theory is $\omega$-inconsistent.

In a nutshell, it a sort of "limit inconsistency" because the theory asserts that there is something that is $F$ but at the same time proves that all numbers are "not-$F$".

Assume now that the theory is inconsistent, i.e. that it proves both $P$ and $\lnot P$. By Explosion: $\lnot P \to (P \to Q)$, we have that the theory proves every formulas, and thus it proves the formula $\exists x F(x)$ ans well as all formulas $\lnot F(n)$, being $\omega$-inconsistent.