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 answer is yes.

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 $A(M)$.

The set-theory-to-arithmetic direction is analogous.


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