# Category Archives: Logic

## 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

$B \colon \textrm{FinSet} \to \textrm{BoolAlg}.$

You can think of $B(S)$ 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.

Leave a comment

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 $\mathbb{N}^M$ 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.

1 Comment

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.

1 Comment

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.

Leave a comment

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.

Leave a comment

Filed under Categories, Conversations, Logic

## Simple Sets and the Recursion Theorem

These notes on Simple Sets are a grabbag about the simple sets of recursion theory. If you don’t know what those are, you probably are not interested, but the Wikipedia article is nice and short and gives the basics.

The last result uses the “shiny black box” (see below), which seems like cheating, but isn’t!

I wrote up the notes sometime in the 1980s based on two papers, and on a lecture by Michael Stob at MIT (reporting on joint work with Maas and Shore). They discuss effectively simple sets and promptly simple sets.

2 Comments

Filed under 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.

3 Comments

Filed under Conversations, Peano arithmetic