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
MT 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
MT (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.