Monthly Archives: September 2020

TOC Post 7 Post 8

Posts 7 and 8 developed an extensive comment thread, mainly between Bruce Smith and John Baez. It was hard to follow in that format, so I converted it to a separate webpage.

Topics: (a) Why do standard models of ZF have standard ω’s? (b) Interactions between the Infinity Axiom and the Foundation Axiom (aka Regularity). (c) The compactness theorem. (d) The correspondence between PA and “ZF with infinity negated”: nonstandard numbers vs. ill-founded sets, and the Kaye-Wong paper (cited in post 8).

TOC Post 7 Post 8

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JB: Before we get into any proofs, I’d just like to marvel at Enayat’s Prop. 6, and see if I understand it correctly. I tried to state it in my own words on my own blog:

Every ZF-standard model of PA that is not V-standard is recursively saturated.

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MW: To my mind, the heart of Enayat’s paper is Proposition 6 and Theorem 7, which combine to give Corollary 8.

Proposition 6: Every ZF-standard model of PA that is nonstandard is recursively saturated.

Theorem 7: Every countable recursively saturated model of PA+Φ_T is a T-standard model of PA.

Corollary 8: The following statements are equivalent for a countable nonstandard model A of arithmetic:

1. A is a T-standard model of PA.
2. A is a recursively saturated model of PA+Φ_T.

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MW: I’d like to chew a bit more on this matter of True_d versus True. This Janus-feature of the Tarski legacy fascinated me from the start, though I didn’t find it paradoxical. But now I’m getting an inkling of how it seems to you.

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