The Jensen Covering Property
نویسندگان
چکیده
The Jensen covering lemma says that either L has a club class of indiscernibles, or else, for every uncountable set A of ordinals, there is a set B ∈ L with A ⊆ B and card(B) = card(A). One might hope to extend Jensen’s covering lemma to richer weasels, that is, to inner models of the form L[ ~ E] where ~ E is a coherent sequence of extenders of the kind studied in Mitchell-Steel [MiSt]. The papers [MiSt], [St3], and [SchSt1] show how to construct weasels with Woodin cardinals and more. But, as Prikry forcing shows, one cannot expect too direct a generalization of Jensen’s covering lemma to weasels with measurable cardinals. Recall from [MiSt] that if L[ ~ E] is a weasel and α is an ordinal, then either Eα = ∅, or else Eα is an extender over J ~ E α . Let us say that L[ ~ E] is a lower-part weasel iff for every ordinal α, Eα is not a total extender over L[ ~ E]. In other words, if L[ ~ E] is a lower-part weasel, then no cardinal in L[ ~ E] is measurable as witnessed by an extender on ~ E. But a lower-part weasel L[ ~ E] could be rich in the sense that it may have levels J ~ E α satisfying ZFC + “there are many Woodin cardinal”. We do not impose any bounds on the large cardinal axioms true in the levels of a lower-part weasel. In this paper we show that if L[ ~ E] is an iterable, lower-part weasel, then either L[ ~ E] has indiscernibles, or else L[ ~ E] satisfies the Jensen covering property. The iterability and indiscernibility that we mean will be made precise in due course. Our result says, in a new way, that Prikry forcing is the essential limitation on extensions of Jensen’s covering lemma.
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ورودعنوان ژورنال:
- J. Symb. Log.
دوره 66 شماره
صفحات -
تاریخ انتشار 2001