Definable MAD families and forcing axioms

Fischer, Vera; Schrittesser, David; Weinert, Thilo

doi:10.1016/j.apal.2020.102909

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…In addition, we say that such a family A is maximal if for every unbounded subset x of , there exists with the property that is unbounded in . Motivated by a classical result of Mathias in [39] that shows that all analytic maximal almost disjoint families in are finite and many additional influential results on maximal almost disjoint families by Mathias, A. Miller, Törnquist, Horowitz and Shelah, Neeman and Norwood, Bakke-Haga, Fischer, Schrittesser, Weinert, and others (see [3, 12, 23, 39, 40, 42, 45, 50]), we will use the techniques developed in the proof of Theorem 1.1 to prove that, if a cardinal possesses sufficiently strong large cardinal properties, then every simply definable almost disjoint family in has cardinality at most . In particular, by a simple diagonalization argument, all simply definable maximal almost disjoint families in have cardinality less than in this case.…”

Section: Introductionmentioning

confidence: 99%

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Lücke,

Müller

2023

Forum of Mathematics, Sigma

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Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$ .

show abstract

Section: Introductionmentioning

confidence: 99%

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Lücke,

Müller

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…In addition, we say that such a family A is maximal if for every unbounded subset x of κ, there exists y ∈ A with the property that x ∩ y is unbounded in κ. Motivated by a classical result of Mathias in [34] that shows that all analytic maximal almost disjoint families in P(ω) are finite and many additional influential results on maximal almost disjoint families by Mathias, A. Miller, Törnquist, Horowitz and Shelah, Neeman and Norwood, Bakke-Haga, Fischer, Schrittesser, Weinert, and others (see [3,11,19,34,35,37,39,44]), we will use the techniques developed in the proof of Theorem 1.1 to prove that, if a cardinal κ possesses sufficiently strong large cardinal properties, then every simply definable almost disjoint family in P(κ) has cardinality at most κ. In particular, by a simple diagonalization argument, all simply definable maximal almost disjoint families in P(κ) have cardinality less than κ in this case.…”

Section: Introductionmentioning

confidence: 99%

$Σ_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Lücke¹,

Müller²

2021

Preprint

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Given an uncountable cardinal κ, we consider the question whether subsets of the power set of κ that are usually constructed with the help of the Axiom of Choice are definable by Σ 1 -formulas that only use the cardinal κ and sets of hereditary cardinality less than κ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical non-definability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of κ of length at least κ + implies the existence of a projective wellordering of the reals. In addition, we determine the exact consistency strength of the non-existence of Σ 1 -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at ω 1 .

show abstract

Definable MAD families and forcing axioms

Cited by 2 publications

References 19 publications

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

$Σ_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

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