MW: Recap: we showed that PAT implies ΦT, where ΦT is the set of all formulas
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.
Category Archives: Conversations
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
You can think of 
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 
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
