By John R. Steel
Huge cardinal hypotheses play a significant function in glossy set idea. One very important strategy to comprehend such hypotheses is to build concrete, minimum universes, or "core models", fulfilling them. considering the fact that Gödel's pioneering paintings at the universe of constructible units, numerous higher middle versions fulfilling more advantageous hypotheses were built, and those have proved relatively necessary. right here the writer extends this idea in order that it may possibly produce center versions pleasurable "There is a Woodin cardinal", a wide cardinal speculation that is the point of interest of a lot present examine. The publication is meant for complicated graduate scholars and reseachers in set conception.
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Extra resources for The Core Model Iterability Problem
As we define the JV α 's we define κa for a < Ω. κa will be the αth infinite cardinal of NΩ . We shall have Na Π (AC Λ + 1) = Nβ Π (Acα + 1) for all β > a. We also maintain inductively that Nβ has the hull property at κ α , for all β > α. Base step: No = W, KQ = ω. Kχ = least AC £ AΓ^ such that Limit step: Kβ < K for all /? < λ . (By induction, AC^ is a cardinal of N\ for all β < λ. ) Successor step: Suppose we are given Na and Ac α , where Na \= κa is a cardinal. For each A C κa such that A £ Na and Λ is a counterexample to Na having the hull property at Ac α , pick a thick class ΓA witnessing this.
Sketch: If Kc \= there is a Woodin cardinal, then its coherent sequence is of size < Ω. Let (T,£/) be a terminal coiteration of Kc with W, using an iteration strategy on the W side and picking unique cofinal branches on the Kc side. (T,U) cannot be successful, since otherwise the Kc side would have iterated past W ', contrary + w + to (a ) = α for stationary many a. Thus it must be that T has no cofinal wellfounded branch. 14: (Kc)ylG1 \= there are no Woodin cardinals. We are not sure whether The Core Model Iterability Problem 41 Ω is (j4o)F-thick in (KC)V^G^ however.
As b is unique, this means that 6 G M[G\[H]. 13. Suppose K(S) is defined, as witnessed by a weasel W such that W |= there are no Woodin cardinals. Let G be V- generic for HP, where P G VΩ. Then V[G] \= Ί<(S) is defined, as witnessed by W* , and K(S)VW = V K(S) . 14. Suppose Kc \= there are no Woodin cardinals, and let G be V -generic over IP G VΩ. e. a < Ω" . Let us observe in passing that if there is an Ω + 1 iterable weasel W such that Ω is 5-thick in W, for some 5, and W \= there are no Woodin cardinals, then in fact Kc |= there are no Woodin cardinals.