# Is it possible to show that this set of L-sentences in the structure $(\omega, +, \cdot, S, <, 0)$ is decidable?

Suppose that $$\Sigma$$ is a consistent set of L-sentences such that there is an L-formula $$\phi$$ such that for all L-sentences $$\psi$$,

$$\Sigma \vdash \psi \iff \phi(\ulcorner\psi\urcorner)$$.

Is it possible to show that $$\Sigma$$ is decidable?