**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 for the ω of *M* equipped with addition and multiplication, defined in the usual way as operations on finite ordinals. So is what he calls a *T*-standard model of PA.

# Category Archives: Conversations

## Non-standard Models of Arithmetic 11

Filed under Conversations, Peano arithmetic

## Non-Standard Models of Arithmetic 10

**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.

Filed under Conversations, Peano arithmetic

## First-Order Categorical Logic 2

**MW:** So let’s see. Last time we talked about the functor *B* from the category FinSet to the category BoolAlg of boolean algebras. Liberal infusions of coffee convinced you that *B* is *covariant*; I accidentally suggested it was *contravariant*. I think I’ve come round to your position, but I still have a couple of things I want to say on the matter. If it won’t be too confusing for our readers.

**JB:** Okay. If we’re planning to talk more about the variance, it’s probably *good *to start out by getting the reader a bit confused. I used to always be confused about it myself. Then I finally felt I had it all straightened out. Then you shocked me by arguing that it worked the opposite way. Your argument was very sneaky.

Filed under Categories, Conversations, Logic

## First-Order Categorical Logic 1

(A conversation between John Baez and Michael Weiss.)

**JB:** Okay, maybe it’s a good time for me to unleash some of my crazy thoughts about logic. They’ve been refined a lot recently, thanks to all the education I’ve been getting from you and folks on the *n*-Category Café. So, I can actually start with stuff that’s not crazy at all… although it may seem crazy if you’re not used to it.

I’ll start with some generalities about first-order classical logic. (I don’t want to get into higher-order logic or intuitionistic logic here!) The first idea is this. In the traditional approach, syntax and semantics start out living in different worlds. In categorical logic, we merge those worlds.

Filed under Categories, Conversations, Logic

## 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.

Filed under Conversations, Peano arithmetic

## 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!

Filed under Conversations, Peano arithmetic

## 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?”

Filed under Conversations, Peano arithmetic

## 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.

Filed under Conversations, Peano arithmetic

## 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 “.

Filed under Conversations, Peano arithmetic

## Non-standard Models of Arithmetic 4

**MW: **I wrote: “I don’t like calling the omega of a model of ZF a *standard model*, for philosophical reasons I won’t get into.”

**JB:** I like it, because I don’t like the idea of “the” standard model of arithmetic, so I’m happy to see that “the” turned into an “a”.

Filed under Conversations, Peano arithmetic