**MW:** I made up a little chart to help me keep all these adjoints straight:

# Category Archives: Conversations

## First-Order Categorical Logic 3

**JB:** Okay, let’s talk more about how to do first-order classical logic using some category theory. We’ve already got the scaffolding set up: we’re looking at functors

You can think of as a set of predicates whose free variables are chosen from the set *S*. The fact that *B* is a functor captures our ability to substitute variables, or in other words rename them.

But now we want to get existential and universal quantifiers into the game. And we do this using a great idea of Lawvere: *quantifiers are adjoints to substitution*.

Filed under Categories, Conversations, Logic

## 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 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.

Filed under Conversations, Peano arithmetic

## 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.

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 beteween 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