Post #2264409

2026-05-06 12:09 UTC

@papalex@mathstodon.xyz @typeswitch@gamedev.lgbt - while I like IST, the quote refers to ZFC and PA, so I was expecting whoever wrote it to be able to give a nonstandard model of PA in a nonstandard model of ZFC which is the unique "initial" model of PA. That's what they seemed claim exists!

Replies (2)

@papalex@mathstodon.xyz 2026-05-06 12:26

@johncarlosbaez@mathstodon.xyz @typeswitch@gamedev.lgbt I must admit that I am not an expert and might have misunderstood the assumptions of the quote. In particular, I read the ZFC part in the quote more as an example rather than an assumption. Though I also don't know prior to googling whether "non-standard model of set theory" implicitly excludes IST? Either way, I would enjoy to learn more about any of these things and am happy to accept if my answer was missing the point, if that is the case? Ignoring any nitty gritty interpretations of the original quote. You wrote "in a non-standard model of ZFA". What actually does this include? I would assume anything with ZFA axioms plus whatever other non-standard axioms goes? EDIT: Ah, I think I get it.. Sorry.. IST is a different theory right? Not a different model?

Open ##2264410
@typeswitch@gamedev.lgbt 2026-05-06 14:28

@johncarlosbaez@mathstodon.xyz @papalex@mathstodon.xyz I'm also not an expert but I'll try to share my intuition here and I would be happy if an expert jumps in with a real example and/or debunking. To make sense of the quote we need a nonstandard model of set theory where the notion of "standard set" makes sense. Presumably, standard sets are sets coming from a standard model of set theory V, which our nonstandard model is built around. So when we say "standard model of PA" we mean the naturals coming from V. 1/

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