Phil 312: Intermediate Logic. Using Stone-Duality. Alejandro Naranjo Sandoval, [email protected] http://scholar.princeton.edu/ansandoval/classes/phi-312-intermediate-logic Some students have expressed some puzzlement about how to use the Stone-Duality theorem to decide the equivalence or non-equivalence of different theories. You can find here a brief guideline meant to alleviate this puzzlement. • Some weeks ago, we proved that the category Th of theories is equivalent to the cate- gory Bool of Boolean Algebras (intuitively speaking, both categories are structurally the same). The Stone-Duality Theorem shows that the categories Bool and Stone are duals of each other (i.e., they are structurally the same, once you reverse the arrows of one of them). One can show that the equivalence of categories is transitive (although strictly speaking we haven't shown this in lecture. But think: you can compose func- tors between categories!). It follows, then, that the categories Th and Stone are also duals of each other (recall from PSet 3 that if you have a translation from a theory T to another theory T 0, then you have a function in the opposite direction from the models of T 0 to the models of T ). • In particular, this means that to each theory T there corresponds a Stone Space. We can call this Stone Space Mod(T ), since it is the space of models S(L(T )), where L(T ) is the usual Lindenbaum Algebra of propositions of T . (Also, given a Stone Space X we can construct a theory that corresponds to it, but this side of the correspondence will be less interesting to us here). • Consider two theories T and T 0. Then we have two Stone Spaces corresponding to each of these, i.e., Mod(T ) and Mod(T 0). This is the crucial point: T and T 0 will be isomorphic in Th iff Mod(T ) and Mod(T 0) are isomorphic in Stone (because both categories are duals of each other). But rememeber: isomorphism in Th means that both theories are homotopy equivalent; and isomorphism in Stone means that both Stone Spaces are homeomorphic to each other (i.e., there is a continuous bijection from one to the other which has a continuous inverse). So, finally, we get: T and T 0 are homotopy equivalent iff Mod(T ) and Mod(T 0) are homeomorphic. • So to show that two theories are not equivalent it suffices to show that their corre- sponding Stone Spaces are not homeomorphic. This is why studying topology is so helpful to us: it allows us to identify properties of spaces which are invariant under homeomorphism, i.e., properties which, if had by a topological space, then any topolog- ical space homeomorphic to it also has. (Often, these properties are called topological properties). Examples of topological properties include compactness, connectedness, having n isolated points, etc. • For example, if one has shown that having a certain number n of isolated points is a topological property, and that Mod(T ) has n isolated points but Mod(T 0) doesn't, then one has also shown that T and T 0 are not homotopy equivalent. 1.