Extending empty set + adjunction to interpret PA

Question

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.)

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

Answer 1

I think it helps to reason semantically here.

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 $\mathsf{Q}$.