MW: OK! So, we’re trying to show that M, the downward closure of B in N, is a structure for L(PA). In other words, M is closed under successor, plus, and times. I’m going to say, M is a supercut of N. The term cut means an initial segment closed under successor (although some authors use it just to mean initial segment).
Monthly Archives: October 2020
MW: So we have our setup: B⊆M⊆N, with N a model of PA, B a set of “diagonal indiscernibles” (whatever those are) in N, and M the downward closure of B in N. So B is cofinal in M, and M is an initial segment of N. I think we’re not going to go over the proof line by line; instead, we’ll zero in on interesting aspects. Where do you want to start?
This is a “reference” post. With all the posts already filed under Peano Arithmetic, I realize I never explicitly stated the axioms. Of course you can find them on Wikipedia and at a large (but finite) number of other places, but I thought I should put them down somewhere on this site.
Last time we looked at Tarski’s inductive definition of truth, expressed informally. We saw how for models of PA, it can be formalized as an infinite sequence of formulas True0, True1, …, formulas belonging to L(PA) itself. But not as a single formula in L(PA).
In post 15 of the Conversation, I observed:
- 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.
The second bullet has occupied its share of pixels in the Conversation. Time for a summing up.
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.
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: