Definable MAD families and forcing axioms

Vera Fischer; David Schrittesser; Thilo Weinert

doi:10.1016/j.apal.2020.102909

Definable MAD families and forcing axioms

Vera Fischer, David Schrittesser, Thilo Weinert

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1 Scopus citations

Abstract

We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π₂¹ infinite MAD families” implies that ω₁ is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π₂¹ infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ₁-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ₃¹ infinite MAD families.

Original language	English
Article number	102909
Journal	Annals of Pure and Applied Logic
Volume	172
Issue number	5
DOIs	https://doi.org/10.1016/j.apal.2020.102909
State	Published - 1 May 2021
Externally published	Yes

Keywords

Bounded proper forcing axiom
MAD families
Maximal almost disjoint families
Remarkable cardinals

ASJC Scopus subject areas

Logic

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10.1016/j.apal.2020.102909

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title = "Definable MAD families and forcing axioms",

abstract = "We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ31 infinite MAD families.",

keywords = "Bounded proper forcing axiom, MAD families, Maximal almost disjoint families, Remarkable cardinals",

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AU - Schrittesser, David

AU - Weinert, Thilo

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N2 - We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ31 infinite MAD families.

AB - We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ31 infinite MAD families.

KW - Bounded proper forcing axiom

KW - MAD families

KW - Maximal almost disjoint families

KW - Remarkable cardinals

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ER -

Definable MAD families and forcing axioms

Abstract

Keywords

ASJC Scopus subject areas

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