# Unpublished works of Woodin on SCH and Radin forcing

There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many unpublished results.

Does any one know some of them to state here.

Is there any way to get a copy of Woodin's unpublished results?

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Why not just ask him? His web page (math.berkeley.edu/~woodin) has his email address. – Nate Eldredge May 30 '13 at 12:34
What is a $T$-degree? (I presume it's not a Turing degree?) – Noah Schweber May 30 '13 at 18:11
(I've moved and expanded my comments to an answer.) – Andrés Caicedo May 31 '13 at 6:24

(Below, I refer to the Handbook. This is the Handbook of Set Theory, Foreman, Kanamori, eds., Springer, 2010.)

The bulk of these results appears in notes by James Cummings. You may want to ask him about them. Originally these notes were intended for a book on Radin forcing to be coauthored by him and Hugh, but significant portions of it appear in his Handbook article. (The book exists in draft form.)

There are a few additional facts about Radin forcing that did not make it into the article, but James has write ups of them, and of course the theory has expanded since.

For $\mathsf{SCH}$ specifically, some details appear in Gitik's paper The negation of the singular cardinal hypothesis from $o(\kappa)=\kappa^{++}$, APAL 43 (1989), 209-234. The state of the art in this regard is described in Gitik's Handbook article, his papers, and those of his co-authors, particularly Carmy Merimovich.

Some applications of Radin/Prikry-like forcing in the context of determinacy are not in any of the above. You can see some in the Koellner-Woodin Handbook article, and yet others in the proofs of the derived model theorem, most of which can be seen (perhaps in preliminary form) here.

The one thing I do not think is in either place is a discussion of $T$-degrees, or constructibility degrees. Hugh discusses some of this (briefly) in The cardinals below $|[\omega_1]^{<\omega_1}|$, APAL 140 (2006), 161–232, but there is a bit more than this.

Richard Ketchersid wrote up the basics in a nice article, More structural consequences of $\mathsf{AD}$, in Set Theory and Its Applications, Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 71-106. To define these degrees, assume the axiom of determinacy, so we have Martin's cone measure on Turing-invariant sets of reals. Given sets of ordinals $T$ and $S$, define $S\lt T$ iff for almost every $r$ (in the sense of Martin's measure) $$L[T,x]\cap\mathbb R\setminus L[S,x]\ne\emptyset.$$ It turns out that for any two sets of ordinals $S,T$, precisely one of the following holds:

1. $S\lt T$, in which case for almost every $x$, the reals of $L[S,x]$ are in $L[T,x]$.
2. $T\lt S$, in which case for almost every $x$, $\mathbb R\cap L[T,x]\subset L[S,x]$.
3. For almost every $x$, $\mathbb R\cap L[T,x]=\mathbb R\cap L[S,x]$.

If option 3 holds, we say that $S$ and $T$ have the same degree. The relation $\lt$ induces a well-ordering of degrees. This is established, together with the basic properties of these degrees, via Prikry-like arguments.

It is possible that there are yet additional results not in any of the above. In that case, I doubt there are formal written notes, but there may be accounts in notes from seminar talks.

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