Definable MAD families and forcing axioms

Vera Fischer; David Schrittesser; Thilo Volker Weinert

doi:10.1016/j.apal.2020.102909

Definable MAD families and forcing axioms

Vera Fischer
, David Schrittesser
, Thilo Volker Weinert

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π ₂ ¹ infinite MAD families” implies that ω ₁ is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π ₂ ¹ infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ ₁-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ ₃ ¹ infinite MAD families.

Original language	English
Article number	102909
Number of pages	16
Journal	Annals of Pure and Applied Logic
Volume	172
Issue number	5
Early online date	2020
DOIs	https://doi.org/10.1016/j.apal.2020.102909
Publication status	Published - May 2021

Austrian Fields of Science 2012

101013 Mathematical logic

Keywords

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

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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 Π 2 1 infinite MAD families” implies that ω 1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π 2 1 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ 1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ 3 1 infinite MAD families. ",

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T1 - Definable MAD families and forcing axioms

AU - Fischer, Vera

AU - Schrittesser, David

AU - Weinert, Thilo Volker

PY - 2021/5

Y1 - 2021/5

N2 - We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π 2 1 infinite MAD families” implies that ω 1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π 2 1 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ 1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ 3 1 infinite MAD families.

AB - We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π 2 1 infinite MAD families” implies that ω 1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π 2 1 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ 1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ 3 1 infinite MAD families.

KW - Bounded proper forcing axiom

KW - MAD families

KW - Maximal almost disjoint families

KW - Remarkable cardinals

UR - https://www.scopus.com/pages/publications/85094173175

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DO - 10.1016/j.apal.2020.102909

M3 - Article

SN - 0168-0072

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JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 5

M1 - 102909

ER -

Definable MAD families and forcing axioms

Abstract

Austrian Fields of Science 2012

Keywords

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