Post #2264424
2026-05-07 13:50 UTC
@papalex@mathstodon.xyz @johncarlosbaez@mathstodon.xyz
I think the problem is confusing. There are three levels here: Peano Arithmetic (the theory), models of PA which live in ZFC set theory (the metatheory), and models of ZFC set theory which live in some ambient set/class theory (the meta-meta-theory). The "standard model of PA" is both a meta-theoretic idea that is internal to ZFC set theory (usually the initial model), and a meta-meta-theoretic idea that relates different models of set theory (N in V and its images).
Replies (1)
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@typeswitch@gamedev.lgbt 2026-05-07 14:00
@papalex@mathstodon.xyz @johncarlosbaez@mathstodon.xyz In both interpretations, 'N in *V is the standard model of PA. This 'N may look different externally but it looks like N in the ZFC set theory (the meta-theory). Where things get interesting is when we have a separate notion of "standard" set in the meta-theory (i.e. if the meta-theory is IST instead of ZFC), and then a meta-theoretic notion of "standard model" means "the initial model that is standard" as opposed to just "the initial model".