116c- Lecture 17

We verified that the sets L_\alpha form a continuous increasing sequence and are transitive. It follows that the reflection theorem holds for the L_\alpha and L. Arguing in \mathsf{ZF}, we proved that L is a model of \mathsf{ZF}, and the reflection theorem allowed us to simplify the proof in a few points.

We then proceeded to argue that L is also a model of choice. In fact, there is a globally definable well-ordering of L. It is worth emphasizing that the well-ordering is a very natural one, as we simply proceed to enumerate the sets in L in the order in which their membership is verified. The definitions of the sequence of sets L_\alpha and of this well-ordering are absolute, and we used this to prove that L is a model of the statement “V=L,” and so is any L_\alpha, for \alpha limit. Moreover, the well-ordering of L, when restricted to L_\alpha, coincides with its interpretation inside L_\alpha.

An easy induction shows that for \alpha infinite, |L_\alpha|=|\alpha|. An argument using the Mostowski collapsing theorem allowed us to prove Gödel’s condensation lemma: If X\prec L_\alpha for \alpha a limit ordinal, then X is isomorphic to some L_\beta. These two facts combine to provide a proof that \mathsf{GCH} holds in L.

Remark. These arguments prove that \mathrm{Con}(\mathsf{ZF}) implies \mathrm{Con}(\mathsf{ZFC}), but they also indicate that showing that \mathrm{Con}(\mathsf{ZF}) implies \mathrm{Con}(\mathsf{ZF}+\lnot\mathsf{AC}) ought to be more complicated. The reason is that the absoluteness of the construction of L implies that if M is a transitive proper class model of \mathsf{ZF}, then L\subseteq M and in fact L^M=L, i.e., the result of running the construction of L from the point of view of M is L itself. But, since V=L holds in L, we cannot prove in \mathsf{ZF} that there is a non-constructible set. If we tried to establish the consistency of \mathsf{ZF} with the negation of choice by a similar method, namely, the construction of a transitive class model M of \mathsf{ZF}+\lnot\mathsf{AC}, then running the construction inside L would give us that L=L^{M^L}\subseteq M^L\subseteq L, so M^L=L, which would be a contradiction, since we are assuming that (provably in \mathsf{ZF}) M is a model of \lnot\mathsf{AC} but L is a model of choice.

This also suggests that in order to show that \mathsf{AC} is independent of \mathsf{ZF}, one should try first to show that V\ne L is consistent with \mathsf{ZF}. The remarkable solution found by Paul Cohen in 1963, the method of forcing, allows us to prove the consistency of both statements, and also to do this while working with transitive models. The method of forcing is beyond the scope of this course, but good explanations can be found in a few places, there is for example a book by Cohen himself, or look at Kunen’s book mentioned at the beginning of the course. Richard Zach has compiled in his blog a list of papers providing an introduction to the method (search for `forcing’).


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