**MW:** Indicators: we don’t *need* to discuss these, to prove the Paris-Harrington theorem. But I think they offer valuable insight.

# Category Archives: Conversations

## Nonstandard Models of Arithmetic 24

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 23

**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).

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## Non-standard Models of Arithmetic 22

**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?

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## 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

## 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.

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## 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).

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## 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.

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## Non-standard Models of Arithmetic 18

**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+Φis a_{T}T-standard model of PA.

Corollary 8:The following statements are equivalent for a countable nonstandard modelAof arithmetic:

Ais aT-standard model of PA.Ais a recursively saturated model of PA+Φ._{T}

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## Non-standard Models of Arithmetic 17

**MW:** I’d like to chew a bit more on this matter of True* _{d}* 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.

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## Non-standard Models of Arithmetic 16

**MW:** Ok, let’s plunge into the construction of True* _{d}*(

*x*). The bedrock level: True

_{0}(

*x*), truth for (closed) atomic formulas. Continue reading

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