By W. Hugh Woodin
This can be the revised and up to date moment variation of a well-established learn monograph at the axiom of determinacy, written via a professional within the box. This axiom is a primary assertion in set idea, and it really is on the topic of profitable thoughts in video game concept.
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Additional resources for The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Second Edition
37. 36 which concerns P<ı . Fix P 2 and 2 Ord. Let ı be a Woodin cardinal such that < ı and such that P 2 Vı . Let D be the set of d Â P such that d is dense in P . 3) there is a ﬁlter F Â X such that F is X -generic. V˛ /. Fix a function H W V˛
N˛C1 is the induced embedding. 1 W N0 ! 1 . 1 . 2 /. 14. 2 -saturated. 1 / Â M . Suppose M # exists. M /. Here MX is the transitive collapse of X . Proof. M /: It sufﬁces to ﬁnd a countable elementary substructure X such that MX is iterable. Fix a cardinal such that M 2 V and such that V M such that X 2 S and ZFC : Thus M 2 V . Let Y V be a countable elementary substructure with M 2 Y and such that Y \ M 2 S . Let X D Y \ M . We claim that MX is iterable.
5) The assertion that a countable transitive model M is iterable is a …12 statement about M and therefore is absolute. (6) Suppose M is iterable and N M is an elementary substructure then in general N may not be iterable. This will follow from results later in this section. In fact here are two natural conjectures. a) Suppose there is no transitive inner model of ZFC containing the ordinals with a Woodin cardinal. Suppose M is a countable transitive model of ZFC and that M is iterable. Suppose X M .
The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Second Edition by W. Hugh Woodin