Some applications of iterated ultrapowers in set theory

Kunen, Kenneth

doi:10.1016/0003-4843(70)90013-6

Cited by 224 publications

(121 citation statements)

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“…[1978]) brought into play the strongest hypothesis to date for establishing a consistency result about the low levels of the cumulative hierarchy. Kunen [1970] had established that having a κ-complete κ + -saturated ideal over a successor cardinal κ has consistency strength substantially stronger than having a measurable cardinal. Kunen now showed: If κ is huge, then there is forcing extension in which κ = ω 1 and there is an ℵ 1 -complete ℵ 2 -saturated ideal over ω 1 .…”

Section: Elaborationsmentioning

confidence: 99%

Large Cardinals with Forcing

Kanamori¹

2012

Handbook of the History of Logic

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Section: Elaborationsmentioning

confidence: 99%

Large Cardinals with Forcing

Kanamori¹

2012

Handbook of the History of Logic

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“…In [3], S. Baldwin generalized Mitchell's results of [16] and showed that it is consistent, relative to measurable cardinals of high Mitchell order, for there to be exactly δ many normal measures on the least measurable cardinal κ, where δ < κ is an arbitrary finite or infinite cardinal. Note that the result of [13] uses forcing, while the results of [12], [16], and [3] use inner model techniques, so that the GCH holds in the models constructed.…”

Section: Introductionmentioning

confidence: 99%

Large cardinals with few measures

Apter

Cummings

Hamkins

2007

Proc. Amer. Math. Soc.

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Abstract. We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly κ + many normal measures on the least measurable cardinal κ. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of λ strong compactness or λ supercompactness measures on P κ (λ) can be exactly λ + if λ > κ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

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“…First, define a ρ-model to be a transitive M containing all the ordinals such that M satisfies ZFC plus the statement ρ is measurable and the universe (i.e., M ) equals L [u], where u is a normal ultrafilter on ρ. By [10], this M is unique (if it exists), so call it M ρ . Furthermore, by [14], M ρ |= GCH, and, by [10], in M ρ , the normal ultrafilter u is unique.…”

mentioning

confidence: 99%

“…By [10], this M is unique (if it exists), so call it M ρ . Furthermore, by [14], M ρ |= GCH, and, by [10], in M ρ , the normal ultrafilter u is unique. Now, u is a subset of the power set P(ρ), but u can be uniquely coded by a set of ordinals.…”

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confidence: 99%

On subclasses of weak Asplund spaces

Kalenda

Kunen

2004

Proc. Amer. Math. Soc.

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Abstract. Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, X, Y , where X is a weak Asplund space such that X * (in the weak* topology) in not in Stegall's class, whereas Y * is in Stegall's class but is not weak* fragmentable.A Banach space X is weak Asplund if every convex continuous real function on X is Gâteaux differentiable at all points of a dense G δ set. This is a large class of spaces; it contains for example all Asplund spaces and all weakly compactly generated spaces. A detailed study of weak Asplund spaces and their subclasses is given in [1]. The largest known subclass of Asplund spaces with reasonable stability properties was introduced by C. Stegall [16].A topological space T belongs to Stegall's class if for any Baire space B and any upper-semicontinuous nonempty compact-valued mapping ϕ : B → T which is minimal with respect to inclusion, ϕ(b) is a singleton for all b ∈ B except for a meager set. If (X * , w * ) is in Stegall's class, then X is weak Asplund. The converse does not hold by [7].A further subclass is the class of those X such that (X * , w * ) is fragmentable. Recall that a topological space T is fragmentable if there exists a metric ρ on the set T such that every nonempty subset of T has nonempty relatively open subsets of arbitrarily small ρ-diameter. Every fragmentable topological space is easily seen to belong to Stegall's class. An example of a Banach space X such that (X * , w * ) is in Stegall's class but not fragmentable is given in [9].Both results of [9] and [7] use some additional axioms beyond ZFC. As remarked in [7], these two sets of axioms cannot hold at the same time. In the present paper, we construct a model of ZFC in which all three classes -weak Asplund spaces, spaces with dual in Stegall's class, and spaces with weak* fragmentable dual -are mutually different. The main result is contained in the following theorem.

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Some applications of iterated ultrapowers in set theory

Cited by 224 publications

References 4 publications

Large Cardinals with Forcing

Large Cardinals with Forcing

Large cardinals with few measures

On subclasses of weak Asplund spaces

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