# Category Archives: Peano Arithmetic

## Nonstandard Models of Arithmetic 26

**MW: **Continuing the recap… Continue reading

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## Nonstandard Models of Arithmetic 25

**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

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

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

**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

## Topics in Nonstandard Arithmetic 7: Truth (Part 3)

Previous “Truth” post Next “Truth” post

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

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.

Filed under Peano Arithmetic