Bounded forcing axioms as principles of generic absoluteness

Bagaria, Joan

doi:10.1007/s001530050154

Cited by 61 publications

(138 citation statements)

References 2 publications

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“…Indeed, in [1] it is shown that for any class of partial orderings P, MA(P) implies Σ We shall prove the following:…”

Section: Lemma 24 Suppose That G Is P(∆ Y )-Generic Over V and Letmentioning

confidence: 92%

“…Fact 1.1 (see [2]). Given ∆ = (δ n ) n > 0, there is 0 < Γ < ∆/2 decreasing which satisfies the following: 1 , x (1) 2 ] or 0, and so on. It is also required that if y n and y m are vectors played by II and n < m, then y n < y m .…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

“…To define the second class of partial orderings we need the notion of ∆-cover: Given s and ∆ = (δ n ) n > 0, we say that a set {t 1 …”

Section: The Ordering Is Given By: (S A) ≤ (T B) Iff T Is a Subsequmentioning

confidence: 99%

“…(x, a 1 , ..., a m ) where a 1 , ..., a m are reals and b is the fixed Borel isomorphism between B 1 and N . For simplicity of notation, suppose that m = 1 and write a for a 1 . Let a be a real that codes both A and a.…”

Section: Working In V [G]mentioning

confidence: 99%

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Determinacy and weakly Ramsey sets in Banach spaces

Bagaria

Lopez-Abad

2001

Trans. Amer. Math. Soc.

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Abstract. We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of c 0 we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper "Weakly Ramsey sets in Banach spaces."

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“…Indeed, in [1] it is shown that for any class of partial orderings P, MA(P) implies Σ We shall prove the following:…”

Section: Lemma 24 Suppose That G Is P(∆ Y )-Generic Over V and Letmentioning

confidence: 92%

Section: Definitions and Basic Factsmentioning

confidence: 99%

“…To define the second class of partial orderings we need the notion of ∆-cover: Given s and ∆ = (δ n ) n > 0, we say that a set {t 1 …”

Section: The Ordering Is Given By: (S A) ≤ (T B) Iff T Is a Subsequmentioning

confidence: 99%

Section: Working In V [G]mentioning

confidence: 99%

See 2 more Smart Citations

Determinacy and weakly Ramsey sets in Banach spaces

Bagaria

Lopez-Abad

2001

Trans. Amer. Math. Soc.

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“…19 For an introductory overview of some of these see Bagaria (2000). 20 That is, the union of all sets whose transitive closure has cardinality less then κ.…”

Section: Appendix: Absoluteness Axiomsmentioning

confidence: 99%

The Search for New Axioms in the Hyperuniverse Programme

Friedman

Ternullo

2016

Boston Studies in the Philosophy and History of Science

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Abstract. The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the 'maximal iterative concept', and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (H-axioms) in countable transitive models, the collection of which constitutes the 'hyperuniverse' (H), has remarkable first-order consequences, some of which we review in §5.

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Bounded Namba forcing axiom may fail

Zapletal

2018

Mathematical Logic Qtrly

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Bounded forcing axioms as principles of generic absoluteness

Cited by 61 publications

References 2 publications

Determinacy and weakly Ramsey sets in Banach spaces

Determinacy and weakly Ramsey sets in Banach spaces

The Search for New Axioms in the Hyperuniverse Programme

Bounded Namba forcing axiom may fail

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