Inhaltsangabe
In these notes we construct an inner model with a Woodin cardinal, and develop fine structure theory for this model. Our model is of the form L[E], where E is a coherent sequence of extenders, and our work builds upon the existing theory of such models. In particular, we rely upon the fine structure theory of L[E] models with strong cardinals, which is due to Jensen, Solovay, Dodd-Jensen, and Mitchell, and upon the theory of iteration trees and "backgrounded" L[EJ models with Woodin cardinals, which is due to Martin and Steel. Our work is what results when fine structure meets iteration trees. One of our motivations was the desire to remove the severe limitations on the theory developed in [MS] caused by its use of an external comparison process. Because of this defect, the internal theory ofthe model L[E] constructed in [MS] is to a large extent a mystery. For example it is open whether the L[EJ of [MS] satisfies GCH. Moreover, the use of an external comparison process blocks the natural generalization to models with infinitely many Woodin cardinals of even the result [MS] does prove about L[E], that L[E] F= CH + ~ has a definable wellordering. Our strategy for making the comparison process internal is due to Mitchell and actually predates [MS]. The strategy includes taking finely calibrated partial ultrapowers ("dropping to a mouse") at certain stages in the comparison process.
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Reseña del editor
In these notes we construct an inner model with a Woodin cardinal, and develop fine structure theory for this model. Our model is of the form L[E], where E is a coherent sequence of extenders, and our work builds upon the existing theory of such models. In particular, we rely upon the fine structure theory of L[E] models with strong cardinals, which is due to Jensen, Solovay, Dodd-Jensen, and Mitchell, and upon the theory of iteration trees and "backgrounded" L[EJ models with Woodin cardinals, which is due to Martin and Steel. Our work is what results when fine structure meets iteration trees. One of our motivations was the desire to remove the severe limitations on the theory developed in [MS] caused by its use of an external comparison process. Because of this defect, the internal theory ofthe model L[E] constructed in [MS] is to a large extent a mystery. For example it is open whether the L[EJ of [MS] satisfies GCH. Moreover, the use of an external comparison process blocks the natural generalization to models with infinitely many Woodin cardinals of even the result [MS] does prove about L[E], that L[E] F= CH + ~ has a definable wellordering. Our strategy for making the comparison process internal is due to Mitchell and actually predates [MS]. The strategy includes taking finely calibrated partial ultrapowers ("dropping to a mouse") at certain stages in the comparison process.
Reseña del editor
This monograph lays the foundations for the theory of canonical inner models of set theory which are large enough to satisfy the statement "There is a Woodin cardinal". It does so by combining Jensen's fine structure models, already useful in the study of smaller inner models, with the theory of iteration trees and Woodin cardinals developed recently by Martin and Steel. The resulting theory is a powerful tool in studying the structure of models of set theory. The main result in this monograph is the construction, given the existence of a Woodin cardinal, of an L-like inner model containing a Woodin cardinal and satisfying the generalized continuum hypothesis, but its real significance is as an indispensable tool for further work with large cardinals in set theory.
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