MW: I’d like to chew a bit more on this matter of Trued versus True. This Janus-feature of the Tarski legacy fascinated me from the start, though I didn’t find it paradoxical. But now I’m getting an inkling of how it seems to you.
Category Archives: Conversations
Non-standard Models of Arithmetic 16
MW: Ok, let’s plunge into the construction of Trued(x). The bedrock level: True0(x), truth for (closed) atomic formulas. Continue reading
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 15
The Truth about Truth
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.
Non-standard Models of Arithmetic 14
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.
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.)
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.)
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.
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?
First-Order Categorical Logic 4
MW: I made up a little chart to help me keep all these adjoints straight:
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.