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anhinga_draftsI'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.
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.
