"Locales and toposes as spaces": models as points

[anhinga_drafts]

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.