Just to check my understanding I have the following (possibly dumb) question: can we build multiple models of PA (one standard and multiple non-standard) within one same model of ZFC?


1 Answer 1


Yes. Being (hopefully helpfully!) very pedantic, the compactness theorem - and its consequence that $\mathsf{PA}$ has nonstandard models - is provable in $\mathsf{ZFC}$, so any model $M$ of $\mathsf{ZFC}$ contains lots of wildly different models of $\mathsf{PA}$. Remember, "provable in $\mathsf{ZFC}$" is exactly the same as "true in every model of $\mathsf{ZFC}$," so as soon as you're convinced that a piece of mathematics is provable in $\mathsf{ZFC}$ (such as the existence of nonstandard models of $\mathsf{PA}$) you know it "occurs" in any model of $\mathsf{ZFC}$.

  • $\begingroup$ clear, thank you very much for the answer and additional explanation! $\endgroup$
    – user341
    Commented May 18 at 15:41

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