Let $\displaystyle A_S^*$ be the set of sentences consisting of S1, S2, and all sentences of the form

$\displaystyle \phi(0) \rightarrow \forall v_1(\phi(v_1) \rightarrow \phi(Sv_1)) \rightarrow \forall v_1 \phi(v_1),$

where $\displaystyle \phi$ is a wff (in the language of $\displaystyle N_S$) in which no variable except $\displaystyle v_1$ occurs free. Show that $\displaystyle A_S \subseteq \text{Cn }A_S^*$. Conclude that $\displaystyle \text{Cn }A_S^* = \text{Th }N_S$. (Here $\displaystyle \phi(t)$ is by definition $\displaystyle \phi_t^{v_1}$. The sentence displayed above is called theinduction axiomfor $\displaystyle \phi$.)

The necessary definitions can be found here (especially, page 2, slide 7).

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We show that if $\displaystyle \psi \in A_S$, then $\displaystyle \psi \in \text{Cn }A_S^*$. Therefore, we show each $\displaystyle S1, S2, S3, S4.1, S4.2 \ldots, S4.n $ is a sentence in $\displaystyle \text{Cn }A_S^*$. If $\displaystyle \psi = S1$ (resp. $\displaystyle S2)$, $\displaystyle \psi \in \text{Cn }A_S^*$ by assumption.

S3 asserts that any nonzero number is the successor of somthing.

How do I show that each S3, S4.1, S4.2, ... , and S4.n is a sentence in $\displaystyle \text{Cn }A_S^*$?

From there, how do I conclude $\displaystyle \text{Cn }A_S^* = \text{Th }N_S$?