MW: OK, let’s recap the setup: we have a three-decker ωU⊂U⊂V. So far as U is concerned, ωU is the “real, true omega”. V knows it isn’t. Enayat’s question: what properties must an omega have, for it to be the omega of a model of T? Here T is a recursively axiomatizable extension of ZF, and U is a model of it.
Category Archives: Peano Arithmetic
Last time I defined ∃n and ∀n prefixes and formulas; Σn, Πn , and Δn relations (and functions) on ℕ; Σn(PA), Πn(PA), and Δn(PA) formulas in L(PA); and Σn(N), Πn(N), and Δn(N) relations (and functions) on a model N of PA. I won’t repeat all that, but a few bullet points may help load it into working memory:
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).
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+ΦT is a T-standard model of PA.
Corollary 8: The following statements are equivalent for a countable nonstandard model A of arithmetic:
- A is a T-standard model of PA.
- A is a recursively saturated model of PA+ΦT.
The Truth about Truth
MW: A little while back, I noted something delicious about the history of mathematical logic:
- Gödel’s two most famous results are the completeness theorem and the incompleteness theorem.
- Tarski’s two most famous results are the undefinability of truth and the definition of truth.