Maximal almost disjoint families, determinacy, and forcing
Copyright policy )Maximal almost disjoint families, determinacy, and forcing
We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy [Formula: see text][Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing.
- Harbin Institute of Technology China (People's Republic of)
- University of Vienna Austria
- University of Copenhagen Denmark
- University of Toronto Canada
- University of Copenhagen Denmark
Determinacy principles, 03E05, 03E15, 03E45, 03E60, Mathematics - Logic, Definability, Other combinatorial set theory, definability, Fubini product, Inner models, including constructibility, ordinal definability, and core models, Mathias forcing, FOS: Mathematics, Borel ideals, determinacy, Logic (math.LO), Descriptive set theory, maximal almost disjoint families
Determinacy principles, 03E05, 03E15, 03E45, 03E60, Mathematics - Logic, Definability, Other combinatorial set theory, definability, Fubini product, Inner models, including constructibility, ordinal definability, and core models, Mathias forcing, FOS: Mathematics, Borel ideals, determinacy, Logic (math.LO), Descriptive set theory, maximal almost disjoint families
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