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Definable MAD families and forcing axioms
Annals of Pure and Applied Logic ( IF 0.6 ) Pub Date : 2020-10-13 , DOI: 10.1016/j.apal.2020.102909
Vera Fischer , David Schrittesser , Thilo Weinert

We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ31 infinite MAD families.



中文翻译:

可定义的MAD族和强迫公理

我们证明 零碳燃料 + BPFA (即有界的正确强迫公理)+“没有 Π21个 无限的MAD家族”意味着 ω1个L的一位重要枢机。换句话说,在BPFA 和反大基数假设有一个 Π21个无限的MAD家庭。因此,一致性强度零碳燃料 + BPFA +“没有投影的无限MAD族”恰好是 Σ1个反射红衣主教高于杰出的红衣主教。相比之下,如果每个实数都具有锐度,则BMM—没有 Σ31个 无限的MAD家庭。

更新日期:2020-10-13
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