# Category Archives: Conversations

## Non-standard Models of Arithmetic 15

MW: A little while back, I noted something delicious about the history of mathematical logic:

• Gödel’s two most famous results are the completeness theorem and the incompleteness theorem.
• Tarski’s two most famous results are the undefinability of truth and the definition of truth.

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 14

MW: Recap: we showed that PAT implies ΦT, where ΦT is the set of all formulas $\{\varphi\rightarrow\text{Con}(T_n+\varphi^\mathbb{N}):\varphi\in\text{L(PA)},n\in\omega\}$

Now we have to show the converse, that PA+ΦT  implies PAT. But first let’s wave our hands, hopefully shaking off some intuition, like a dog shaking off water.

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 13

MW: OK, back to the main plotline. Enayat asks for a “natural” axiomatization of PAT. Personally, I don’t find PAT all that “unnatural”, but he needs this for Theorem 7. (It’s been a while, so remember that Enayat’s T is a recursively axiomatizable extension of ZF.)

Filed under Conversations, Peano Arithmetic

## First-Order Categorical Logic 6

MW: An addendum to the last post. I do have an employment opportunity for one of those pathological scaffolds: the one where B(0) is the 2-element boolean algebra, and all the B(n)’s with n>0 are trivial. It’s perfect for the semantics of a structure with an empty domain.

The empty structure has a vexed history in model theory. Traditionally, authors excluded it from the get-go, but more recently some have rescued it from the outer darkness. (Two data points: Hodges’ A Shorter Model Theory allows it, but Marker’s Model Theory: An Introduction forbids it.)

Filed under Categories, Conversations, Logic

## First-Order Categorical Logic 5

JB: Okay, let me try to sketch out a more categorical approach to Gödel’s completeness theorem for first-order theories. First, I’ll take it for granted that we can express this result as the model existence theorem: a theory in first-order logic has a model if it is consistent. From this we can easily get the usual formulation: if a sentence holds in all models of a theory, it is provable in that theory.

Filed under Categories, Conversations, Logic

## Non-standard Models of Arithmetic 12

JB: It’s been a long time since Part 11, so let me remind myself what we’re talking about in Enayat’s paper Standard models of arithmetic.

We’ve got a theory T that’s a recursively axiomatizable extension of ZF. We can define the ‘standard model’ of PA in any model of T, and we call this a ‘T-standard model’ of PA. Then, we let PAT to be all the closed formulas in the language of Peano arithmetic that hold in all T-standard models.

This is what Enayat wants to study: the stuff about arithmetic that’s true in all T-standard models of the natural numbers. So what does he do first?

Filed under Conversations, Peano Arithmetic

## First-Order Categorical Logic 4

MW: I made up a little chart to help me keep all these adjoints straight: Filed under Categories, Conversations, 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.

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.