Category Archives: Peano Arithmetic

Topics in Nonstandard Arithmetic 10: Truth (Part 4)

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Nonstandard Models of Arithmetic 26

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MW: Continuing the recap… Continue reading

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Nonstandard Models of Arithmetic 25

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MW: It’s been ages since John Baez and I discussed Enayat’s paper—not since October 2020! John has since moved on to fresh woods and pastures new. I’ve been reading novels. But I feel I owe it to our millions of readers to finish the tale, so here goes.

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Topics in Nonstandard Arithmetic 9: Tricks with Quantifiers

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Every specialty has its tricks of the trade. They become second nature to practitioners, so they often don’t make it into the textbooks. Quantifiers rule in logic; here are some of the games we can play with them. I’ll start with tricks that apply in logic generally, then turn to those specific to Peano arithmetic.

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Topics in Nonstandard Arithmetic 8: Extensions and Substructures

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Substructures and extensions loom large in math: subgroups, subrings, extension fields, submanifolds, subspaces of topological spaces… So too in the model theory of PA.

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Nonstandard Models of Arithmetic 24

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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

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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

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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?

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Topics in Nonstandard Arithmetic 7: Truth (Part 3)

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Last time we looked at Tarski’s inductive definition of truth formalized inside ZF set theory. Continue reading

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Topics in Nonstandard Arithmetic 6: The Axioms

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This is a “reference” post. With all the posts already filed under Peano Arithmetic, I realize I never explicitly stated the axioms. Of course you can find them on Wikipedia and at a large (but finite) number of other places, but I thought I should put them down somewhere on this site.

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