The PCF conjecture and large cardinals

Pereira, Luís Alexandre

doi:10.2178/jsl/1208359066

Cited by 6 publications

(6 citation statements)

References 23 publications

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“…Pereira shows in [24] that the existence of a continuous tree-like scale on a product n<ω τ n guarantees the failure of AFSP with respect to τ n n (see also Lemma 15), and further proves that continuous tree-like scales, unlike other well-known types of scales, such as good scales, can exist in models with some of the strongest large cardinal notions, e.g. I 0 -cardinals.…”

Section: Introductionmentioning

confidence: 85%

Approachable Free Subsets and Fine Structure Derived Scales

Adolf,

Ben-Neria

2021

Preprint

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Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders {o(µ) | µ < λ} is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

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Section: Introductionmentioning

confidence: 85%

Approachable Free Subsets and Fine Structure Derived Scales

Adolf,

Ben-Neria

2021

Preprint

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“…A crucial concept for the constructions in this paper is a tree-like scale. This concept appears in Shelah [12] (see II Conclusion 3.5) and was isolated and further studied by Pereira in [11].…”

Section: Preliminariesmentioning

confidence: 99%

“…Pereira described a forcing notion in [11] which produces a continuous tree-like scale and preserves cardinals, and hence also the approachability property at κ (a principle which implies that every scale on κ is good) if it holds in the ground model.…”

Section: Preliminariesmentioning

confidence: 99%

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Tight stationarity and tree-like scales

Chen¹

2015

Annals of Pure and Applied Logic

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Let κ be a singular cardinal of countable cofinality, κ n : n < ω be a sequence of regular cardinals which is increasing and cofinal in κ. Using a scale, we define a mapping μ from n P(κ n ) to P(κ + ) which relates tight stationarity on κ to the usual notion of stationarity on κ + . We produce a model where all subsets of κ + are in the range of μ for some κ a singular. Using a version of the diagonal supercompact Prikry forcing, we obtain such a model where κ is strong limit. Then we construct a sequence of stationary sets that is not tightly stationary in a strong way, namely, its image under μ is empty. All of these results start from a model with a continuous tree-like scale on κ.

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“…These are of interest to practitioners of pcf theory for use in the construction of scales -see Pereira [12]. The definition we give is in the spirit of that given in Devlin [4], Chapter 13, although we only require P (κ) to be constructibly coded rather than explicitly listed.…”

Section: Universal Morassesmentioning

confidence: 99%