Maximal almost disjoint families, determinacy, and forcing

Abstract

We study the notion of $\mathcal{J}$-MAD families where $\mathcal{J}$ is a Borel ideal on $\omega$. We show that if $\mathcal{J}$ is an arbitrary $F_{\sigma }$ ideal, or is any finite or countably iterated Fubini product of $F_{\sigma }$ ideals, then there are no analytic infinite $\mathcal{J}$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal{J}$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal{J}$-mad families. These results apply in particular when $\mathcal{J}$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.

40 pages

References in corpus (2)

• Can you take Toernquist's inaccessible away?
• Maximal independent sets in Borel graphs and large cardinals

Cited by in corpus (5)

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• Compact Sets of Baire Class One Functions and Maximal Almost Disjoint Families
• Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH
• The Ramsey property and higher dimensional mad families
• The Ramsey property implies no mad families