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

## 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

## Topics in Nonstandard Arithmetic 1: Table Setting

TOC Next

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

## Non-standard Models of Arithmetic 19

**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

## Bundles and Laplacians

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.