Few new reals

Asperó, David; Mota, Miguel Angel

doi:10.48550/arxiv.1712.07724

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…By Lemma 2.7, we can find U ⊆ ℓg(q) of size ℵ 1 such that C ˜is a P q U -name and by enlarging it, we may assume that U is q-closed. Now by Lemma 2.1, we can guarantee clause (3). Item (4) follows from Lemma 2.8…”

Section: Thus Any P ω 2 -Namementioning

confidence: 91%

“…It is worth to mention that a positive answer to the above question (namely the consistency of measuring with CH) was recently announced by Asperó and Mota [3] using a finite support forcing iteration with a countable symmetric system of models with markers and a finite undirected graph on the symmetric system as side conditions, however their forcing adds new reals, though only ℵ 1 -many reals, and in [3] the following question is asked (the question is attributed to Moore): Question 0.3. ( [3]) Does Measuring imply that there are non-constructible reals. Theorem 0.2 gives a negative answer to this question as well.…”

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confidence: 99%

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The measuring principle and the continuum hypothesis

Golshani¹,

shelah²

2022

Preprint

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We show that one can force the Measuring principle without adding any new reals. We also show that it is consistent with the large continuum. These results answer two famous questions of Justin Moore. § 0. IntroductionIn this paper we study Moore's measuring principle and prove two consistency results related to it. We show that one can force it by a proper forcing notion which adds no new reals. We also show that it is consistent with the continuum being arbitrary large. These results show that the measuring principle has no effects on the size of the continuum.Before we continue, let us start by recalling the definition of the measuring principle. Definition 0.1. Measuring holds iff for every sequence C = C δ : δ < ω 1 , δ limit , if each C δ is a club of δ, then there is a club C ⊆ ω 1 which measures C, i.e., for every δ ∈ C, there is some α < δ such that either

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Section: Thus Any P ω 2 -Namementioning

confidence: 91%

mentioning

confidence: 99%

The measuring principle and the continuum hypothesis

Golshani¹,

shelah²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…To our knowledge, the first such model was constructed by Jensen [DJ74]. However, a simpler construction is possible using recent work of Aspero and Mota [AM17]. Do these models separate the above two principles for some H?…”

Section: IImentioning

confidence: 99%

The open dihypergraph dichotomy for generalized Baire spaces and its applications

Schlicht¹,

Sziráki²

2023

Preprint

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The open graph dichotomy for a subset X of the Baire space ω ω states that any open graph on X either admits a coloring in countably many colors or contains a perfect complete subgraph. It is a strong version of the open coloring axiom for X that was introduced by Todorčević and Feng to study definable sets of reals. We first show that its recent infinite dimensional generalization by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay's model, extending a theorem of Feng from dimension 2. Our main theorem lifts this result to generalized Baire spaces κ κ in two ways.(1) For any regular infinite cardinal κ, the following holds after a Lévy collapse of an inaccessible cardinal λ > κ to κ + . Suppose that H is a κ-dimensional box-open directed hypergraph on a subset of κ κ such that H is definable from a κ-sequence of ordinals. Then either H admits a coloring in κ many colors or there exists a continuous homomorphism from a canonical large directed hypergraph to H. (2) If λ is a Mahlo cardinal, then the previous extends to all relatively box-open directed hypergraphs on any subset of κ κ that is definable from a κ-sequence of ordinals. We derive several applications to definable subsets of generalized Baire spaces, among them variants of the Hurewicz dichotomy that characterizes subsets of Kσ sets, an asymmetric version of the Baire property, an analogue of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an Fσ set, the determinacy of Väänänen's perfect set game for all subsets of κ κ, and an analogue of the Jayne-Rogers theorem that characterizes the functions which are σ-continuous with closed pieces. Some of these applications lift results of Carroy, Miller and Soukup from the countable setting and extend results of Väänänen, Lücke, Motto Ros and the authors in the uncountable setting.

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Few new reals

Cited by 2 publications

References 16 publications

The measuring principle and the continuum hypothesis

The measuring principle and the continuum hypothesis

The open dihypergraph dichotomy for generalized Baire spaces and its applications

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