Maximal almost disjoint families, determinacy, and forcing

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We study the notion of -MAD families where is a Borel ideal on ω. We show that if is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite -MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective -MAD families; and under the full Axiom of Determinacy + V = L(a) or under AD+ there are no infinite-mad families. Similar results are obtained in Solovay's model. These results apply in particular to the ideal Fin, which corresponds to the classical notion of MAD families, as well as to the ideal Fin. The proofs combine ideas from invariant descriptive set theory and forcing.

OriginalsprogEngelsk
Artikelnummer2150026
TidsskriftJournal of Mathematical Logic
Vol/bind22
Udgave nummer1
Sider (fra-til)1-42
ISSN0219-0613
DOI
StatusUdgivet - 2022

ID: 276955168