Category Archives: Peano arithmetic

Non-standard Models of Arithmetic 11

MW: Time to start on Enayat’s paper in earnest. First let’s review his notation. M is a model of T, a recursively axiomatizable extension of ZF. He writes \mathbb{N}^M for the ω of M equipped with addition and multiplication, defined in the usual way as operations on finite ordinals. So \mathbb{N}^M is what he calls a T-standard model of PA.

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Non-Standard Models of Arithmetic 10

(MW: I have converted the first few posts into pdf files, formatted both for a small screen screen and a medium-sized one.)

JB: So, last time you sketched the proof of the Paris–Harrington theorem. Your description is packed with interesting ideas, which will take me a long time to absorb. Someday I should ask some questions about them. But for now I’d like to revert to an earlier theme: how questions about the universe of sets cast their shadows down on the world of Peano arithmetic.

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Non-standard Models of Arithmetic 9

MW: Time to talk about the Paris-Harrington theorem. Originally I thought I’d give a “broad strokes” proof, but then I remembered what you once wrote: keep it fun, not a textbook. Anyway, Katz and Reimann do a nice job for someone who wants to dive into the details, without signing up for a full-bore grad course in model theory. So I’ll say a bit about the “cast of characters” (i.e., central ideas), and why I think they merit our attention.

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Non-standard Models of Arithmetic 8

JB: So, you were going to tell me a bit how questions about the universe of sets cast their shadows down on the world of Peano arithmetic.

MW: Yup. There are few ways to approach this. Mainly I want to get to the Paris-Harrington theorem, which Enayat name-checks.

First though I should do some table setting of my own. There’s a really succinct way to compare ZF with PA: PA = ZF − infinity!

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Non-standard Models of Arithmetic 7

MW: Our goal for the next few posts is to understand Enayat’s paper

• Ali Enayat, Standard models of arithmetic.

JB: Yee-hah!

MW: I’m going to take a leisurely approach, with “day trips” to nearby attractions (or Sehenswürdigkeiten, in the delightful German phrase), but still trying not to miss our return flight.

Also, I know you know a lot of this stuff. But unless we’re the only two reading this (in which case, why not just email?), I won’t worry about what you know. I’ll just pretend I’m explaining it to a younger version of myself—the one who often murmured, “Future MW, just what does this mean?”

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Non-standard Models of Arithmetic 6

JB: It’s interesting to see how you deploy various philosophies of mathematics: Platonism, intuitionism, formalism, etc. For a long time I’ve been disgusted by how people set up battles between these, like Punch-and-Judy shows where little puppets whack each other, instead of trying to clarify what any of these philosophies might actually mean.

For example, some like to whack Platonism for its claim that numbers “really exist”, without investigating what it might mean for an abstraction—a Platonic form—to “really exist”. If you define “really exist” in such a way that abstractions don’t do this, that’s fine—but it doesn’t mean you’ve defeated Platonism, it merely means you’re committed to a different way of thinking and talking.

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Non-standard Models of Arithmetic 5

MW: John, you wrote:

Roughly, my dream is to show that “the” standard model is a much more nebulous notion than many seem to believe.

and you gave a good elucidation in post 2 and post 4. But I’d like to defend my right to “true arithmetic” and “the standard model \mathbb{N}“.

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