JB: So, let’s think about how we can prove this generalization of Gödel’s completeness theorem. First, remember that a hyperdoctrine B is consistent iff B(0) has at least two elements, or in other words, ⊤ ≠ ⊥ in this boolean algebra. Second, let’s say a hyperdoctrine C is set-based if every C(n) is the power set of Vn for some fixed set V. We call V the universe. Third, let’s say a morphism of hyperdoctrines, say F: B → C, is a natural transformation whose components F(n): B(n) → C(n) are Boolean algebra homomorphisms obeying the Beck–Chevalley condition and maybe the Frobenius condition. (We’re a bit fuzzy about this and we’ll probably have to sharpen it up.) Continue reading
Category Archives: Categories
First-Order Categorical Logic 13
MW: It’s been a minute! Well, almost 60,000 minutes.
We left off with a question: does a natural transformation from a syntactic hyperdoctrine to a semantic hyperdoctrine automatically “respect quantifiers”? We saw that this amounts to a Beck–Chevalley condition. We wondered if we had to add that condition to our definition of a model, or if it came for free. Continue reading
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First-Order Categorical Logic 12
MW: Last time we looked at the categorical rendition of “C is a model of B”:
- Functors B:FinSet→BoolAlg and C:FinSet→BoolAlg
- A natural transformation F:B→C
where B and C are hyperdoctrines, and
- B is syntactic: the elements of each B(n) are equivalence classes of formulas (which we agreed to call predicates);
- C is semantic: the elements of each C(n) are relations on a domain V.
(We’ve been saying that C(n) is the set of all n-ary relations on V, but I see no need to assume that.)
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First-Order Categorical Logic 11
MW: Last time we justified some equations and inequalities for our adjoints: they preserve some boolean operations, and “half-preserve” some others. And we incidentally made good use of the color palette!
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First-Order Categorical Logic 10
JB: Last time we saw how to get some laws of logic from two facts:
• right adjoint functors between boolean algebras preserve products (‘and’),
and
• left adjoint functors between boolean algebras preserve coproducts (‘or’).
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First-Order Categorical Logic 9
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First-Order Categorical Logic 8
MW: We’re reviewing hyperdoctrines, which are specially nice functors B: FinSet → BoolAlg. When we have such a functor, any map f of finite sets gives a homomorphism of boolean algebras, B(f). But we’ve seen this is a morphism and a functor. (“It’s a floor wax and a dessert topping!”) What do you think about the term “adjoint morphism”? It might help keep the two levels straight.
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First-Order Categorical Logic 7
MW: John, it’s been eons since we last discussed First-Order Categorical Logic: not since September 2019! (I read a lot of Russian novels during the break.) But New Year’s seems like a good time to resume the tale.
JB: Yes indeed! It’s been a long time, and it’s mostly my fault. Let’s see if we can get back up to speed.
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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.)
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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.
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Diagonal Argument 