**MW:** Last time we learned about the “back-and-forth” condition for two countable structures *M* and *N* for a (countable) language *L*:

# Category Archives: Conversations

## Nonstandard Models of Arithmetic 31

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 30

**MW:** Time to finish off Enayat’s Theorem 7:

**Theorem 7:** Every countable recursively saturated model *N* of PA+Φ* _{T}* is a

*T*-standard model of PA.

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 29

**MW:** We’re still going through Enayat’s proof of his Theorem 7:

**Theorem 7:** Every countable recursively saturated model *N* of PA+Φ* _{T}* is a

*T*-standard model of PA.

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 28

**MW:** I ended the last post with a puzzle. Here it is again, in more detail.

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 27

**MW:** Enayat’s second major result is:

**Theorem 7:** Every countable recursively saturated model of PA+Φ* _{T}* is a

*T*-standard model of PA.

Filed under Conversations, Peano Arithmetic

## Nonstandard Models of Arithmetic 24

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

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).

Filed under Conversations, Peano Arithmetic

## 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?

Filed under Conversations, 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

## 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