On PA and ZFC-Inf

[JOSEPH SHIPMAN]

It is sometimes said that PA is equivalent in strength to ZF-Inf, or to ZF-Inf+~Inf.

I want to understand the distinction between these.

Is there any sentence P in the language of set theory, for which (assuming ZF is consistent) exactly one of the following is two statements is true?

1) ZF-Inf proves that P is true in the structure V_omega
2) ZF-Inf+~Inf proves P 

— JS

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