Some applications of the core model

Donder, Dieter; Jensen, Ronald Björn; Koppelberg, Bernd

doi:10.1007/bfb0098619

Cited by 45 publications

(34 citation statements)

References 8 publications

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“…In general, C(ω 1 ) ≤ E(ω 1 ). Silver [4] proved that the consistency of ZFC + "E(ω 1 ) exists" implies that of ZFC + C(ω 1 ) = ω 2 , and Donder [2] proved the reverse relative consistency.…”

Section: Proposition 3 Let L[e] Be An Extender Model With the Propertmentioning

confidence: 99%

Collapsing functions

Schimmerling¹,

Veličković

2003

Mathematical Logic Qtrly

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We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for (2 ω 1 ) + , then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L [E]. The least cardinal λ with the Chang property (λ, ω1) (ω1, ω) is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j(ω1) is the least ω1-Erdös cardinal.One of the striking consequences of large cardinals is that they imply the existence of a generic elementary embedding j : V −→ M with M transitive and crit(j) = ω 1 . For example, if δ is a Woodin cardinal, then there is a condition in the non-stationary tower P <δ that forces the existence of such an embedding (see [6]). The value of j(ω 1 ) tends to be rather large. For example, if δ is a Woodin cardinal in an iterable extender model L[E], then forcing with P L[E] <δ over L[E] produces an embedding j with j(ω 1 ) ≥ the least ω 1 -Erdös cardinal of L[E].It is natural to ask if large cardinals imply the existence of a precipitous ideal on ω 1 since this would imply the existence of a generic elementary embedding j with j(ω 1 ) < (2 ω1 ) + . One way to disprove this might be to show that there is a set forcing which kills all precipitous ideals on ω 1 . In this paper we present some partial results on this question and, in particular, show that Woodin limits of Woodin cardinals do not imply the existence of precipitous ideals on ω 1 .The contents of the paper are as follows. First we define what it means for there to exist a collapsing function for a cardinal λ. Then we show that if there exists a collapsing function for (2 ω1 ) + , then no ideal on ω 1 is precipitous. It is easy, as we show, to add a collapsing function for ω 2 by forcing; whether this can be done for ω 3 is not known.Next we turn to inner models of the form L [E], where E is a coherent sequence of extenders. We define what it means for a cardinal to be weakly ω 1 -Erdös. Then we show that in L[E], there is a collapsing function on λ if and only if λ is less than the least weakly ω 1 -Erdös cardinal. In particular, in L [E], there are no precipitous ideas on ω 1 . As a corollary to this and an earlier result of Itay Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of a precipitous ideal on ω 1 .We conjecture that in L[E], there is a precipitous ideal on κ if and only if κ is measurable. John Steel [5] has shown this by a method different from ours under hypotheses more rest...

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Section: Proposition 3 Let L[e] Be An Extender Model With the Propertmentioning

confidence: 99%

Collapsing functions

Schimmerling¹,

Veličković

2003

Mathematical Logic Qtrly

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“…We note that the scale involved in the principle CS is a "Very Good Scale" in the sense of our paper [4]. This principle is closely related to some combinatorial principles of Donder et al [8] and Donder et al [7]. Definition 3.1.…”

Section: A Non-tight Mutually Stationary Sequencementioning

confidence: 95%

Canonical structure in the universe of set theory: part two

Cummings¹,

Foreman

Magidor

2006

Annals of Pure and Applied Logic

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We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 (1-3) (2004) 211-243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness.

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“…By the definition, we have that F is a perfect tree, i.e., for each s £ F there are so ± sx such that so < s and sx < s. everywhere. One uses the Jensen JE^ absoluteness [11] instead of Shoenfield JE2 absoluteness.…”

Section: Open Coloring Of Pairs From Coanalytic Setsmentioning

confidence: 99%

Homogeneity for Open Partitions of Pairs of Reals

Qi¹

1993

Transactions of the American Mathematical Society

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Abstract. We prove a partition theorem for analytic sets of reals, namely, if A C R is analytic and [A]2 -Kq U Kx with Ko relatively open, then either there is a perfect O-homogeneous subset or A is a countable union of 1-homogeneous subsets. We also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset. A two person game for this partition property is also studied. There are some applications of such partition properties.

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Some applications of the core model

Cited by 45 publications

References 8 publications

Collapsing functions

Collapsing functions

Canonical structure in the universe of set theory: part two

Homogeneity for Open Partitions of Pairs of Reals

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