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

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

An easy induction shows that for infinite, . An argument using the Mostowski collapsing theorem allowed us to prove Gödel’s condensation lemma: If for a limit ordinal, then is isomorphic to some . These two facts combine to provide a proof that holds in .

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

This also suggests that in order to show that is independent of , one should try first to show that is consistent with . 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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