Hi Wei Dai, About books. Concerning the provability logics I always mentionned the Boolos 1993 (or even his lovely lighter Boolos 1979), but I would like to mention also the book "Self-reference and modal logic" by Smorynski. The only problem is its very little caracters; I should go to the occulist! :0 But he has a nice chapter on the algebraic models of the provability logic: the so called diagonalisable algebras and fixed point algebras. As you know the Z logics I got are so weak that they loose (like G*) Kripke semantics or even Scott-Montague (sort of topological) semantics. So we need some algebraic move. Note that the G/G* story *begun* with those diagonalisable algebra through the work of Magari (Italy).
But, perhaps more importantly at this stage I must recall the book "Mathematics of Modality" by Robert Goldblatt. It contains fundamental papers on which my "quantum" derivation relies. I mentionned it a lot some time ago. And now that I speak about Goldblatt, because of Tim May who dares to refer to algebra, category and topos! I want mention that Goldblatt did wrote an excellent introduction to Toposes: "Topoi". (One of the big problem in topos theory is which plural chose for the word "topos". There are two schools: topoi (like Goldblatt), and toposes (like Bar and Wells). :) Goldblatt book on topoi has been heavily attacked by "pure categorically minded algebraist like Johnstone for exemple, because there is a remnant smell of set theory in topoi. That is true, but that really help for an introduction. So, if you want to be introduced to the topos theory, Goldblatt Topoi, North Holland editor 19?(I will look at home) is perhaps the one. -Bruno PS I get your questions. I will think a little bit before answering. Thanks to Tim for Egan's exerp.
