**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?

# Category Archives: Logic

## Non-standard Models of Arithmetic 22

Filed under Conversations, Peano Arithmetic

## Topics in Nonstandard Arithmetic 7: Truth (Part 3)

Previous “Truth” post Next “Truth” post

Last time we looked at Tarski’s inductive definition of truth formalized inside ZF set theory. Continue reading

Filed under Peano Arithmetic

## Topics in Nonstandard Arithmetic 6: The Axioms

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.

Filed under Peano Arithmetic

## Non-standard Models of Arithmetic 21

Bruce Smith joins the conversation, returning to a previous topic: the Paris-Harrington theorem. (Discussion of the Enayat paper will resume soon.)

Filed under Conversations, Peano Arithmetic

## Topics in Nonstandard Arithmetic 5: Truth (Part 2)

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 True_{0}, True_{1}, …, formulas belonging to L(PA) itself. But not as a *single* formula in L(PA).

Filed under Peano Arithmetic

## Topics in Nonstandard Arithmetic 4: Truth (Part 1)

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.

Filed under Peano Arithmetic

## Non-standard Models of Arithmetic 20

**MW:** OK, let’s recap the setup: we have a three-decker ω* ^{U}*⊂

*U*⊂

*V*. So far as

*U*is concerned, ω

*is the “real, true omega”.*

^{U}*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

## Topics in Nonstandard Arithmetic 3: The Arithmetic Hierarchy (Part 2)

Last time I defined ∃* _{n}* and ∀

*prefixes and formulas; Σ*

_{n}*, Π*

_{n}*, and Δ*

_{n}*relations (and functions) on ℕ; Σ*

_{n}*(PA), Π*

_{n}*(PA), and Δ*

_{n}*(PA) formulas in L(PA); and Σ*

_{n}*(*

_{n}*N*), Π

*(*

_{n}*N*), and Δ

*(*

_{n}*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

## Nonstandard Arithmetic: A Long Comment Thread

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