First-Order Categorical Logic 6

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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.)

All n-ary predicates with n>0 are universally true, because they’re vacuously true. But ∃x(x=x) is false, while ∀x(x=x) is true. So B(0) contains both verum ⊤ and falsum ⊥.

The empty structure looks a little sketchy in this sense: the standard rules of logic require modification for it. (That’s why you were able to prove that its logic isn’t a hyperdoctrine.) On the other hand, its good friends the number 0 and the empty set will vouch for its good character. Hodges allows it because it makes some theorems nicer.

JB: This is very interesting. Near the end of our last conversation I claimed without proof that the Frobenius condition implies

x(⊤) = ⊤.

I now think this was wrong. Without this, I don’t think we can rule out the hyperdoctrine where B(0) is the 2-element boolean algebra but all the B(n)’s with n>0 are trivial. And that makes me happy, because I really don’t want to rule out the empty model of first-order theories—that is, what you’re calling the ’empty structure’.

So let me revise some of my statements from the last episode. I want to define an inconsistent hyperdoctine B to be one with ⊤ = ⊥ in B(0), or equivalently, one where B(0) is the trivial boolean algebra. This implies that all the B(n)’s with n>0 are trivial.

If any B(n)’s with n>0 is trivial, then they all are—our techniques from last time do show this—but this does not imply that B(0) is trivial. There is a hyperdoctrine where B(0) is the 2-element boolean algebra but all the B(n)’s with n>0 are trivial. And this one is consistent.

To really prove this we should dig deeper into the Frobenius condition—and for that matter, the Beck-Chevalley condition. If I try to explain these to you, that will force me to understand me better.

But I think we’re okay for now. So, this time it’s for real: on to Gödel’s completeness theorem!

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