Fine structure and iteration trees. (English) Zbl 0805.03042

Mitchell, William J.; Steel, John R.

Lecture Notes in Logic. 3. Berlin: Springer-Verlag. v, 130 p. DM 48.00; öS 374.40; sFr 48.00 /sc (1994).

These notes describe the construction of an inner model with a Woodin cardinal and develop fine structure theory of this model. The process relies on the fine structure of \(L[\vec E]\) models with strong cardinals and on the theory of iteration trees and “backgrounded” \(L[\vec E]\) models with Woodin cardinals. The notes show what happens when fine structure meets iteration trees.
The notes use an internal comparison process. The strategy uses finely calibrated partial ultrapowers at certain stages of the process. The notes can be viewed as a long inductive proof that a certain construction yields a model \(L[\vec E]\) whose levels have certain fine structural properties. Among these is a strong local form of GCH. A corollary of this result is that if ZFC + “There is a Woodin cardinal” is consistent, then so is ZFC + “There is a Woodin cardinal”+ GCH. The main interest however is not in this result, but in the power of the method to decide many other questions about this model and similar models containing more Woodin cardinals.
The work described here has been informally circulated since October 1989. The introduction briefly outlines other results which have been obtained since then.

Reviewer: A.M.Coyne (Asheville)

MSC:

03E45	Inner models, including constructibility, ordinal definability, and core models
03E55	Large cardinals
03-02	Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Keywords:

inner model; Woodin cardinal; fine structure; iteration trees; finely calibrated partial ultrapowers