Is arithmetic on the naturals $\omega$-consistent?

Question

Is there any first order theory of arithmetic of the natural numbers that is known to be $\omega$-consistent if it is consistent?

If yes, then how?

Answer 1

The empty theory over the language of arithmetic is certainly $\omega$-consistent.

This theory has the property that whenever it proves some formula $\phi$, you can replace every atomic formula $t_1=t_2$ by "true" and ignore all of the quantifiers, and the resulting Boolean expression will then evaluate to true. (Intuitively this is because the one-element universe is a model, but you can verify it purely syntactically one inference rule at a time).

Therefore it is impossible for the theory to prove both $\neg\phi(0)$ and $(\exists x)\,\phi(x)$, since the two Boolean expressions they translate to are each other's negations.

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(This same argument also works for full Peano Arithmetic minus the axiom stating that $0$ is not a successor).