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HAPPY AND MAD FAMILIES IN L(ℝ)

Published online by Cambridge University Press:  01 August 2018

ITAY NEEMAN  and
ZACH NORWOOD
ITAY NEEMAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095-1555, USAE-mail:ineeman@math.ucla.edu
ZACH NORWOOD
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095-1555, USAE-mail:znorwood@math.ucla.edu
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Abstract

We prove that, in the choiceless Solovay model, every set of reals is H-Ramsey for every happy family H that also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under $A{D^ + }$.

Keywords

03E3503E55mad familieshappy familiesSolovay modeldeterminacy
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 572 - 597
Copyright
Copyright © The Association for Symbolic Logic 2018 

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