We study the notion of 𝒥-MAD families where 𝒥 is a Borel ideal on ω. We show that if 𝒥 is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite 𝒥-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective 𝒥-MAD families; and under the full Axiom of Determinacy +V=L(ℝ) or under AD+ there are no infinite 𝒥-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⊗Fin. The proofs combine ideas from invariant descriptive set theory and forcing.

中文翻译：

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最大几乎不相交的家庭，确定性和强迫

我们研究的概念𝒥-MAD 家庭𝒥是一个 Borel 理想ω. 我们证明如果𝒥是有限集理想的任何有限或可数迭代 Fubini 乘积鳍，则不存在解析无穷大𝒥-MAD 族，并假设射影确定性和依赖选择没有无限射影𝒥-疯狂的家庭；并在完整的确定性公理下+五=大号(ℝ)或以下一个D+没有无限的𝒥-疯狂的家庭。在 Solovay 的模型中也得到了类似的结果。这些结果尤其适用于理想鳍，它对应于 MAD 家族的经典概念，以及理想的鳍⊗鳍. 证明结合了不变描述集理论和强迫的想法。