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 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.
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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.
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Nonstandard Models of Arithmetic 28
MW: I ended the last post with a puzzle. Here it is again, in more detail.
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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.
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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.
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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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Non-standard Models of Arithmetic 20
Trudy Campbell
MW: OK, let’s recap the setup: we have a three-decker ωU⊂U⊂V. So far as U is concerned, ωU is the “real, true omega”. 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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Diagonal Argument 