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

# Author Archives: Bruce Smith

## 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.)

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