Definable maximal discrete sets in forcing extensions
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Definable maximal discrete sets in forcing extensions. / Schrittesser, David; Törnquist, Asger Dag.
I: Mathematical Research Letters, Bind 25, Nr. 5, 2018, s. 1591-1612.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Definable maximal discrete sets in forcing extensions
AU - Schrittesser, David
AU - Törnquist, Asger Dag
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
U2 - 10.4310/MRL.2018.v25.n5.a11
DO - 10.4310/MRL.2018.v25.n5.a11
M3 - Journal article
VL - 25
SP - 1591
EP - 1612
JO - Mathematical Research Letters
JF - Mathematical Research Letters
SN - 1073-2780
IS - 5
ER -
ID: 184033460