Weak covering without countable closure

Mitchell, William J.; Schimmerling, Ernest

doi:10.4310/mrl.1995.v2.n5.a6

Cited by 40 publications

(59 citation statements)

References 3 publications

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“…Set Λ i = sup(π"λ i ) for i ≤ ϕ. The proof of [MiSc95] now shows that we may and shall assume that the following hold true, for every i ≤ ϕ. Claim 1.…”

Section: (Lemma 34)mentioning

confidence: 99%

“…We may and shall moreover assume that all objects occuring in the proof of [MiSc95] are iterable. Let T be the normal iteration tree on K arising from the coiteration withK.…”

Section: (Lemma 34)mentioning

confidence: 99%

“…be such that (Y ; ...) ≺ M, (κ i | i < δ) ∈ Y , and that all the objects occuring in the proof of [MiSc95] are iterable.…”

Section: Lemma 312mentioning

confidence: 99%

“…We may and shall assume that ((K(X),M),λ) is not iterable. Furthermore, by the method of [MiSc95], we may and shall assume that the phalanxes occuring in the proof to follow are all iterable. Letλ ′ ≤Ω be largest such thatK(X)|λ ′ does not move in the coiteration with M # n+1 (X).…”

Section: (Theorem 39)mentioning

confidence: 99%

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PCF theory and Woodin cardinals

Gitik¹,

Schindler²,

Shelah³

2017

Logic Colloquium '02

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Mathematics Subject Classification. Primary 03E04, 03E45. Secondary 03E35, 03E55.Keywords: cardinal arithmetic/pcf theory/core models/large cardinals. AbstractTheorem 1.1. Let α be a limit ordinal. Suppose that 2 |α| < ℵ α and 2 |α| + < ℵ |α| + , whereas ℵ |α| α > ℵ |α| + . Then for all n < ω and for all bounded The proofs of these theorems combine pcf theory with core model theory. Along the way we establish some ZFC results in cardinal arithmetic, motivated by Silver's theorem [Si74], and we obtain results of core model theory, motivated by the task of building a "stable core model." Both sets of results are of independent interest.

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“…Set Λ i = sup(π"λ i ) for i ≤ ϕ. The proof of [MiSc95] now shows that we may and shall assume that the following hold true, for every i ≤ ϕ. Claim 1.…”

Section: (Lemma 34)mentioning

confidence: 99%

“…We may and shall moreover assume that all objects occuring in the proof of [MiSc95] are iterable. Let T be the normal iteration tree on K arising from the coiteration withK.…”

Section: (Lemma 34)mentioning

confidence: 99%

“…be such that (Y ; ...) ≺ M, (κ i | i < δ) ∈ Y , and that all the objects occuring in the proof of [MiSc95] are iterable.…”

Section: Lemma 312mentioning

confidence: 99%

Section: (Theorem 39)mentioning

confidence: 99%

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PCF theory and Woodin cardinals

Gitik¹,

Schindler²,

Shelah³

2017

Logic Colloquium '02

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“…Here we apply a frequent extension argument, similar to that in Schimmerling-Steel [SS99]; see also Mitchell-Schimmerling [MS95]. A general version of the frequent extension argument in the simplified context of extender models below 0 ¶ is described in Zeman [Zem02, § §7.5, 8.3], and also Cox [Cox09].…”

Section: Proof Of Theorem 6(a)mentioning

confidence: 99%

Downward Transference of Mice and Universality of Local Core Models

Caicedo¹,

Zeman²

2017

J. symb. log.

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Abstract. If M is a proper class inner model of ZFC and ω M 2 = ω2, then every sound mouse projecting to ω and not past 0 ¶ belongs to M. In fact, under the assumption that 0 ¶ does not belong to M, K M ω2 is universal for all countable mice in V.Similarly, if M is a proper class inner model of ZFC, δ > ω1 is regular, (δ + ) M = δ + , and in V there is no proper class inner model with a Woodin cardinal, then K M δ is universal for all mice in V of cardinality less than δ.

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