Let me begin with a couple of comments that may help clarify some of the results from last lecture.

First, I want to show a different proof of Lemma 21.2, that I think is cleaner than the argument I gave before. (The argument from last lecture, however, will be useful below, in the proof of Kunen’s theorem.)

**Lemma 1** * If is measurable, is a -complete nonprincipal ultrafilter over and is the corresponding ultrapower embedding, then *

*Proof:* Recall that if is Mostowski’s collapsing function and denotes classes in then To ease notation, write for

Let Pick such that for all

**Lemma 2** * With notation as above, for any *

*Proof:* For a set let denote the function constantly equal to Since is an isomorphism, ‘s lemma gives us that the required equality holds iff

but this last set is just

From the nice representation just showed, we conclude that for all But for any such because by Lemma 21 from last lecture. Hence, which is obviously in being definable from and

The following was shown in the proof of Lemma 20, but it deserves to be isolated.

**Lemma 3** * If is a normal nonprincipal -complete ultrafilter over the measurable cardinal then i.e., we get back when we compute the normal measure derived from the embedding induced by *

Finally, the construction in Lemma 10 and preceeding remarks is a particular case of a much more general result.

**Definition 4** * Given and an ultrafilter over the ***projection** of over is the set of such that

Clearly, is an ultrafilter over

Notice that if is a partition of into sets not in and is given by the unique such that then is a -complete nonprincipal ultrafilter over (Of course, is possible.)

For a different example, let be a -complete nonprincipal ultrafilter over the measurable cardinal and let represent the identity in the ultrapower by Then is the normal ultrafilter over derived from the embedding induced by

**Definition 5** * Given ultrafilters and (not necessarily over the same set), say that is ***Rudin-Keisler below** in symbols, iff there are sets and a function such that

**Theorem 6** * Let be an ultrafilter over a set and an ultrafilter over a set Suppose that Then there is an elementary embedding such that *

*Proof:* Fix and for which there is a map such that Clearly, as witnessed by the map and similarly so it suffices to assume that and

Given let be given by Then is well-defined, elementary, and

In effect, iff iff iff where the second equivalence holds by assumption, and it follows that is well-defined.

If denotes the function with domain and constantly equal to then for any since by definition of the map This shows that

Elementarity is a straightforward modification of the proof of Lemma 10 from last lecture.

One can show that Theorem 6 “very nearly” characterizes the Rudin-Keisler ordering, see for example Proposition 0.3.2 in Jussi Ketonen, *Strong compactness and other cardinal sins*, Annals of Mathematical Logic **5** (1972), 47–76.

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