This is a continuation of my last post on determinacy, where we began the proof of projective determinacy. We’ve reduced the statement to showing that every projective set is homogeneously Suslin, which will be shown here, modulo a key lemma from [MS89].
As previously mentioned we will inductively show that every set is homogeneously Suslin, assuming a measurable above a limit of Woodins. The start of the induction is the following theorem, derived from Martin’s proof of -determinacy from a measurable.
Theorem (Martin, ’70). Suppose is a measurable cardinal. Then every set is -homogeneously Suslin.
I’ll here define the tree and measures that witness this fact, but without showing that it works. For the interested reader, this can be found in [Kanamori, 32.1 & 31.1]. Fix some set . By the characterisation of sets (see [Kanamori, 13.1]) we can find a tree on such that, given any ,
iff is wellfounded. (1)
Fix some recursive enumeration and define for each a strict linear ordering on as iff
or or ,
where is the Kleene-Brouwer ordering of . We then see by (1) that
iff is a wellordering on . (2)
Given any define and furthermore set to be the ordering on defined exactly as but replacing with . Note that and for any . We can now define the tree we’re interested in as
Note here that the second coordinates are providing witnesses for the wellfoundedness of the ordering associated to the first coordinate. By (2) we then see that indeed, . So far so good. As for the measures, letting be a normal measure on , define for each
To see that this works one uses that holds by Rowbottom’s theorem — see details in [Kanamori, 32.1 & 32.2].
Having covered the base case we then want to go from assuming that every set is homogeneously Suslin to the corresponding fact about sets. Firstly, we need to weaken the notion of a homogeneously Suslin set.
Definition(s). A tree over for any set is -weakly homogeneous if there exists a countable set of -complete measures such that for any there is a countably complete towert with for every .
A set is then -weakly homogeneously Suslin if for a -weakly homogeneous tree, and weakly homogeneously Suslin if it’s -weakly homogeneously Suslin for some .
An alternative characterisation of the weakly homogeneously Suslin sets is that is -weakly homogeneously Suslin iff for some -homogeneously Suslin set (see [Kanamori, 32.3]). The following fact is then the crucial connection between the homogeneously Suslin sets and their weak variants.
Key Lemma (Martin-Steel, ’89). Let be a Woodin cardinal and fix some . Then if is -weakly homogeneously Suslin, is -homogeneously Suslin for every .
This is the main theorem in [MS89], and for the purposes of this post we’ll just black box the result. Using this Key Lemma we can now prove projective determinacy.
Theorem (Martin-Steel, ’89). Assume there exist Woodins and a measurable above. Then determinacy holds.
Proof. Let be an increasing enumeration of the Woodins, set and let be the measurable. For each fix some ordinal and set . We will show by induction on that every set is -homogeneously Suslin for every .
For we get the result by the above theorem due to Martin. Assume now that every set is -homogeneously Suslin for some and fix some set . By definition of we can fix some set such that .
By our inductive hypothesis is -homogeneously Suslin, so that is -weakly homogeneously Suslin by the above alternative characterisation of the weakly homogeneously Suslin sets. Since , the Key Lemma implies that is -homogeneously Suslin. As homogeneously Suslin sets are determined, we’re done. QED
So, modulo the Key Lemma, a result I might or might not write up a proof of at some point, we’ve shown projective determinacy from infinitely many Woodins and a measurable above. It turns out that from this hypothesis it’s even possible to prove that all sets of reals in are determined, a tremendous result due to Woodin, which will be covered next time.
- [Kanamori] Kanamori, Akihiro: “The Higher Infinite”, second edition, 2009.
- [MS89] Martin, Donald A. and Steel, John R.: “A Proof of Projective Determinacy”, Journal of the American Mathematical Society, 2(1): 71-125, 1989.