Definable maximal discrete sets in forcing extensions

David Schrittesser; Asger Dag Törnquist

doi:10.4310/MRL.2018.v25.n5.a11

Definable maximal discrete sets in forcing extensions

Institut for Matematiske Fag

Abstract

Let  be a Σ11 binary relation, and recall that a set A is -discrete if no two elements of A are related by . We show that in the Sacks and Miller forcing extensions of L there is a Δ12 maximal -discrete set. We use this to answer in the negative the main question posed in [5] by showing that in the Sacks and Miller extensions there is a Π11 maximal orthogonal family ("mof") of Borel probability measures on Cantor space. A similar result is also obtained for Π11 mad families. By contrast, we show that if there is a Mathias real over L then there are no Σ12 mofs.

Originalsprog	Engelsk
Tidsskrift	Mathematical Research Letters
Vol/bind	25
Udgave nummer	5
Sider (fra-til)	1591-1612
ISSN	1073-2780
DOI	https://doi.org/10.4310/MRL.2018.v25.n5.a11
Status	Udgivet - 2018

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10.4310/MRL.2018.v25.n5.a11

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title = "Definable maximal discrete sets in forcing extensions",

abstract = " Let R be a Σ 1 1 binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ 1 2 maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π 1 1 maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ 1 2 mofs.",

author = "David Schrittesser and T{\"o}rnquist, {Asger Dag}",

year = "2018",

doi = "10.4310/MRL.2018.v25.n5.a11",

language = "English",

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T1 - Definable maximal discrete sets in forcing extensions

AU - Schrittesser, David

AU - Törnquist, Asger Dag

PY - 2018

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N2 - Let R be a Σ 1 1 binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ 1 2 maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π 1 1 maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ 1 2 mofs.

AB - Let R be a Σ 1 1 binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ 1 2 maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π 1 1 maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ 1 2 mofs.

U2 - 10.4310/MRL.2018.v25.n5.a11

DO - 10.4310/MRL.2018.v25.n5.a11

M3 - Journal article

SN - 1073-2780

VL - 25

SP - 1591

EP - 1612

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 5

ER -

Definable maximal discrete sets in forcing extensions

Abstract

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