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- W2000550033 abstract "The full-background-extender K c -construction of [2] has the property that, if it does not break down and produces a final model , then Ή is Woodin in V ⇒ Ή is Woodin in , for all Ή. It is natural to ask whether κ is strong in V ⇒ κ is λ-strong in , for all κ, or even better, κ is λ-strong in V ⇒ κ is λ-strong in . As one might suspect, the more useful answer would be “yes”. For the K c -construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V , then The natural proof only gives that if κ is Σ 2 -strong, then Σ, is strong in . We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that (1) iteration trees on can be lifted to iteration trees on V , (2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and (3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in )." @default.
- W2000550033 created "2016-06-24" @default.
- W2000550033 creator A5088247102 @default.
- W2000550033 date "2007-09-01" @default.
- W2000550033 modified "2023-09-27" @default.
- W2000550033 title "Local Kc constructions" @default.
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- W2000550033 doi "https://doi.org/10.2178/jsl/1191333838" @default.
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