"Locales and toposes as spaces": Lindenbaum algebras

I've now completed the second reading of the Vickers' text I mentioned in the previous posting. I even understood it almost completely, although I would not be able to pass a closed book exam. I would probably do OK on an open book exam.

One cool thing is that Vickers explains on 3 pages (Section 2.1) what the Lindenbaum algebras are good for. I knew the definition, but I did not understand much about it, except that this is yet another way to form a partial order with elements being sets of propositions.

It turns out that (say, for classical propositional logic, where Lindenbaum algebras are Boolean algebras), if we denote the Lindenbaum algebra of a theory T as L(T), then for any Boolean algebra A there is a bijection between Boolean algebras homomorphisms L(T) → A and models of the theory T in A.

That's very cool: instead of talking about models in A you can simply talk about homomorphisms from L(T) to A.

"Locales and toposes as spaces": models as points

Let's look at the bijection mentioned in the previous post more closely. A model is "a map from the syntactic theory to a semantic space". In particular, given a Lindenbaum algebra L(T) we can build a "generic model" of T called M_T in L(T) (by mapping the propositional variables to their classes of equivalences in L(T) ).

Now, for a homomorphism f : L(T) → A, the corresponding model of the theory T in A is a map f o M_T (mapping the propositional variables to the elements of A).

Now, if we look at the post of June 9, we'll see a duality between generalized points and models. "Generic point" and "generic model" correspond to each other, and global points 1 → X correspond to standard models in the 2-valued Boolean algebra 2 = {false, true}, that is to the homomorphisms L(T) → 2.

One of the main ideas in the Vickers' text is that one can think about all kinds of spaces as spaces of models (one takes a space and starts thinking about its points as models of a logical theory of an appropriate type, usually more complex than classical propositional logic), and that in a large variety of logical situations one can take all models and form a space from them.