I’ve now put up a draft of my expository talk on forcing, which I gave a month ago to the math grad students at Berkeley. I’m hoping that it should make basic independence proofs intelligible to math grad students, and to interested philosophers with a fairly technical disposition. If you read it, certainly leave comments here, or e-mail me (easwaran AT berkeley DOT edu), if there are any questions, unclarities, inaccuracies, comments, or anything else. I think the first page has a pretty pompous tone, so I should probably change that, but I’m not sure about the rest.
Anyway, the reason I decided to write it up (apart from being able to explain this material more clearly to people in future discussion and talks), is because it’s seemed to me that there’s no accessible introduction to this stuff available (EDIT: since making this post, I noticed that Ars Mathematica mentioned forcing for dummies by Tim Chow – I’m reading it now). All the set theory books seem to either just do basic stuff with ordinals and cardinals (excluding forcing and large cardinals and determinacy and the like) or put forcing at least 100 or 200 pages in and rely on a lot of the material that’s discussed in earlier chapters. However, I recently discovered (thanks to the latest issue of Phi-News, which I ran into through a post by Gillian Russell) John Bell’s book Set Theory: Boolean-Valued Models and Independence Proofs. This book presupposes some amount of familiarity with set theory, but it just jumps right into this material right away. And I’ve found it much easier to read than the relevant chapters of Jech or Kunen (but perhaps that’s at least in part because I spent so much time in August and September going through the relevant chapters of Jech). So there’s not as much of a hole to fill as I thought, but I’m trying not to presuppose any theorems of set theory beyond Russell’s paradox.
Bell’s book is also quite interesting to me because it presents the material entirely in the framework of boolean-valued models, rather than forcing. The results and the methods are almost entirely equivalent, but the formulation is different. The method of forcing requires the existence of countable transitive models of ZF (which aren’t guaranteed by Con(ZF)), but then gives a standard model-theoretic consistency proof by explicitly creating a model of the new theory. The method of boolean-valued models works on the universe as a whole, rather than on some subset of it. But as a result, it doesn’t actually construct the domain of some structure for the theory – instead it gives a proper class with some boolean-valued (rather than true/false-valued) relations on it representing identity and set membership, and shows that the set of formulas receiving value “1” must be consistent, and can be made to include theories of appropriate sorts. The boolean-valued approach has the advantage of making all the calculations of “truth-values” for sentences much easier, but the disadvantage of making the model somehow “blurry” and indistinct. Forcing, on the other hand, gives a clear model, in exchange for some extra calculational difficulties.
I’ve always felt more comfortable with an approach highlighting boolean-valued models much more than Kunen does, and probably even a bit more than Jech, but Bell’s approach has felt alien by the fact that it doesn’t mention what seems to be the more standard approach at all until page 88. At any rate, it’s been an interesting read so far.