# Is “PA+ω-rule” and “Zermelo-infinity+every set is finite + ω-set-rule” equi-interpretable?

We know that "PA" and "Zermelo-infinity+every set is finite" are equi-interpretable.

Now is "PA+$$\omega$$-rule" and "Zermelo-infinity+every set is finite + $$\omega$$-set-rule" equi-interpretable?

where the $$\omega$$-set-rule is:

$$for \ n=0,1,2,3,... \\ \forall x_1,..,x_n \forall x [\forall y (y \in x \leftrightarrow y=x_1 \lor ..\lor y=x_n) \to \psi(x)]$$

.....

$$\forall k (\psi(k)$$

The usual interpretations in both directions still work. E.g. if $$M$$ is a model of PA, let $$A(M)$$ be the corresponding model of ZF-Inf+Fin gotten from $$M$$ via the Ackermann interpretation. We just check that if $$M$$ satisfies the $$\omega$$-rule then $$A(M)$$ satisfies the $$\omega$$-set rule.
The key point is that we can define cardinality, and so $$(*)_\psi:\quad\mbox{"In the Ackermann interpretation, every n-element set has property \psi"}$$ can be expressed in the language of arithmetic. If $$\psi$$ is an instance of the $$\omega$$-set rule, then $$(*)_\psi$$ is an instance of the $$\omega$$-rule, and since the $$\omega$$-rule holds in $$M$$ we get that $$\forall x\psi(x)$$ holds in $$A(M)$$.