Category Archives: Conversations

Nonstandard Models of Arithmetic 31

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

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+ΦT 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+ΦT 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+ΦT 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: BMN, 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.)