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Trudy Campbell
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.
Filed under Conversations, 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:
Filed under Peano Arithmetic
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).
Filed under Conversations, Peano Arithmetic
John Baez and I have been having a conversation about nonstandard models of Peano arithmetic (PA). It started with “John’s dream”, as I’ll call it, which begat a goal: understand a paper by Enayat. For more on the dream and the goal, check out the second post.
Filed under Peano Arithmetic
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.
Filed under Conversations, Peano Arithmetic
I originally started this blog to make available various notes I’ve written over the years. (Justification: the internet hasn’t yet run out of space.) Herewith a very short note on principal and fiber bundles (small and medium formats), and a longer one on the Laplacian on the cube. Also Three takes on the tangent bundle.
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.
Filed under Conversations, Peano Arithmetic
MW: I’d like to chew a bit more on this matter of Trued 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.
Filed under Conversations, Peano Arithmetic
MW: Ok, let’s plunge into the construction of Trued(x). The bedrock level: True0(x), truth for (closed) atomic formulas. Continue reading
Filed under Conversations, Peano Arithmetic
Diagonal Argument 