# Extending empty set + adjunction to interpret PA

Let N = empty set + adjunction. N interprets Q.1 Q + induction yields PA.

Does N + epsilon-induction interpret PA? If so:

• Are they mutually interpretable, sententially equivalent, and/or bi-interpretable?

• Is there an even simpler X such that N + X interprets PA?

If not: Is there a simple X such that N + X interprets PA?

I leave the notion of "simple" deliberately vague, intending it as some kind of conceptual simplicity. (ZFfin and PA are bi-interpretable, but I seek something simpler than ZFfin.)

1. A minimal predicative set theory. Antonella Mancini, Franco Montagna. Notre Dame Journal of Formal Logic. 35 (2): 186–203. Spring 1994.

Let $$\Phi$$ be the usual interpretation of $$\mathsf{Q}$$ in $$\mathsf{N}$$, and suppose $$\mathcal{M}\models\mathsf{N}+\epsilon\mathsf{Ind}$$. Then we can show that $$\Phi^\mathcal{M}$$ (= the model of $$\mathsf{Q}$$ that $$\Phi$$ "builds from" $$\mathcal{M}$$) also satisfies $$\mathsf{PA}$$: a definable cut in $$\Phi^\mathcal{M}$$ would give us a definable set with no $$\epsilon$$-minimal element in $$\mathcal{M}$$.
So not only does $$\mathsf{N+\epsilon Ind}$$ interpret $$\mathsf{PA}$$, it does so in exactly the same way that $$\mathsf{N}$$ interprets $$\mathsf{Q}$$.