Abstract
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.
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