On the compactness of ℵ<sub>1</sub> and ℵ<sub>2</sub>

Prisco, Carlos Augusto Di; Henle, Jim

doi:10.2307/2273517

Cited by 15 publications

(16 citation statements)

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“…An argument similar in spirit to that of 2.1.12 answers a question of [1]. Namely, assuming ZF + ADR + 0 is regular, one can show that the two supercompactness measures on Pw(co2) defined in [1] are both identical to the measure defined (implicitly) in [8, §3].…”

Section: Proof Suppose P Lh Vx G R 3 Vmentioning

confidence: 91%

Two consequences of determinacy consistent with choice

Steel¹,

Wesep²

1982

Trans. Amer. Math. Soc.

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Abstract. We begin with a ground model satisfying ZF + AD + ACR, and from it construct a generic extension satisfying ZFC + ô2 = w2 + "me nonstationary ideal on w | is w2-saturated". 0. Introduction. The "axiom" of determinacy (henceforth AD) implies many interesting propositions about small cardinal numbers, for example 2, and the second that the nonstationary ideal on w, is w2-saturated. We shall show that the theory consisting of ZFC + ADL(R) together with these two propositions is consistent. (ADL(R) is the assertion that all games in £(R), a large class of definable games, are determined.)In consonance with our working hypothesis, we would like to prove our result assuming only the consistency of ZFC + ADL(R), or equivalently, that of ZF + AD + DC (cf. §1). At present we cannot do this. We instead assume the consistency of ZF + AD + ACR, where ACR is the axiom of choice for families indexed by the reals. Consistency-wise, ZF + AD + ACR is one of the strongest theories known to man; we show in §1 that it proves the consistency of ZF + AD + DC. The only upper bound we know on the consistency strength of ZF + AD + ACR is that of ZF + ADR + "Ö is regular". We offer some partial justification for our use of such a strong hypothesis at the end of the paper.Our consistency proof is by forcing. We start with a ground model M satisfying ZF + AC + ACR. By a two-step iteration we construct a generic extension M [C7] "onto M so that z\h (h: w2 -* R ). The forcing will add no new reals and preserve w2, i.e.,We then pass t0 the inner model N _ £(RM ^ an(J show it has the desired properties. The paper is organized as follows: In §1 we give some preliminaries and background information. In particular, we set forth those consequences of AD whose

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Section: Proof Suppose P Lh Vx G R 3 Vmentioning

confidence: 91%

Two consequences of determinacy consistent with choice

Steel¹,

Wesep²

1982

Trans. Amer. Math. Soc.

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“…Since any cardinal γ which is δ +α supercompact is automatically <δ supercompact, by the remarks immediately following the proof of Lemma 2.1, we know that in V P , δ is a limit of cardinals γ which are <δ supercompact. As V P "δ is δ +α strongly compact", by a theorem of DiPrisco [7], V P "Any cardinal γ which is either <δ supercompact or <δ strongly compact is δ +α strongly compact". Thus, in V P , δ is a limit of cardinals γ which are δ +α strongly compact, a contradiction.…”

Section: Proofmentioning

confidence: 99%

Supercompactness and partial level by level equivalence between strong compactness and strongness

Apter

2004

Fund. Math.

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Abstract.We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if δ +α is regular, δ is δ +α strongly compact iff δ is δ +α+1 strong, except possibly if δ is a limit of cardinals γ which are δ +α strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible. Introduction and preliminaries.In [1], the following theorem was proven.Theorem 1. Let V "ZFC + κ is supercompact + There is no pair of cardinals δ < λ such that δ is λ supercompact and λ is measurable". There is then a partial ordering P ⊆ V such that V P "ZFC + GCH + There is no pair of cardinals δ < λ such that δ is λ supercompact and λ is measurable + κ is both the least strongly compact and least strong cardinal (so κ is notThis theorem provides a counterpoint to the main result of [5], which is as follows.There is then a partial ordering P ⊆ V such that V P "ZFC + GCH + K is the class of supercompact cardinals + For every pair of regular cardinals κ < λ, κ is λ strongly compact iff κ is λ supercompact, except possibly if κ is a limit of cardinals δ which are λ supercompact".2000 Mathematics Subject Classification: 03E35, 03E55. Key words and phrases: supercompact cardinal, strongly compact cardinal, strong cardinal, measurable cardinal, non-reflecting stationary set of ordinals, level by level equivalence between strong compactness and supercompactness, level by level equivalence between strong compactness and strongness.[123]

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“…On the other hand, Solovay showed that if U is a K-complete, normal ultrafilter over PKA, and {/a's over PKa for k < a < A are defined by A E Ua if and only if {y G PKA: y D a E A} E U, then the glued-together ultrafilter over PKA of the Ua's does not have the partition property (see [2]). …”

Section: Claim {X E a : F(x(t])) G X For Every Tj < K} G û Proof Ofmentioning

confidence: 99%

“…These ultrafilters in turn can be "glued together" with a measure on X to again produce an ultrafilter over PKX. In general, the ultrafilter over PKX obtained by this method is not the same one which is started out with (see [2]). In §3 it is shown that if the ultrafilter constructed over PKX in §2 is restricted to ultrafilters over PKa for k < a < X, then glued together with the measure constructed on X in this same section, the glued together ultrafilter is the same as the one started out with.…”

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The ultrafilter characterization of huge cardinals

Mignone¹

1984

Proc. Amer. Math. Soc.

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Abstract. A huge cardinal can be characterized using ultrafilters. After an argument is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice.Introduction. The notion of a huge cardinal was first introduced by Kunen. Its primary definition is given in terms of elementary embeddings. And just as measurable and supercompact cardinals have ultrafilter characterizations in ZFC, a huge cardinal can also be characterized by ultrafilters.In §1, after the elementary embedding definition for a huge cardinal and two equivalent ultrafilter characterizations are given, an argument is made for one characterization above the other in instances where the axiom of choice is not available. Once an ultrafilter characterization for k being huge is settled upon-which defines k as huge if for some X > k there exists a K-complete, normal ultrafilter over {x C X: x = k}-two theorems are proved in §2 which state that providing such an ultrafilter exists over {x czX: x = k}, then X is measurable and k is A-supercompact. These are both well-known theorems in ZFC, but here they are established in ZF without the axiom of choice.Finally, given an ultrafilter over PKX it can be restricted to ultrafilters over PKa for k < a < X. These ultrafilters in turn can be "glued together" with a measure on X to again produce an ultrafilter over PKX. In general, the ultrafilter over PKX obtained by this method is not the same one which is started out with (see [2]). In §3 it is shown that if the ultrafilter constructed over PKX in §2 is restricted to ultrafilters over PKa for k < a < X, then glued together with the measure constructed on X in this same section, the glued together ultrafilter is the same as the one started out with.

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On the compactness of ℵ₁ and ℵ₂

Cited by 15 publications

References 4 publications

Two consequences of determinacy consistent with choice

Two consequences of determinacy consistent with choice

Supercompactness and partial level by level equivalence between strong compactness and strongness

The ultrafilter characterization of huge cardinals

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On the compactness of ℵ1 and ℵ2

Cited by 15 publications

References 4 publications

Two consequences of determinacy consistent with choice

Two consequences of determinacy consistent with choice

Supercompactness and partial level by level equivalence between strong compactness and strongness

The ultrafilter characterization of huge cardinals

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Product

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On the compactness of ℵ₁ and ℵ₂