We show that $\mathsf{ZFC}$ + $\mathsf{BPFA}$ (i.e., the Bounded Proper Forcing Axiom) + “there are no ${\mathbf{\Pi }}_2^1$ infinite MAD families” implies that ${\omega }_1$ is a remarkable cardinal in L. In other words, under $\mathsf{BPFA}$ and an anti-large cardinal assumption there is a ${\mathbf{\Pi }}_2^1$ infinite MAD family. It follows that the consistency strength of $\mathsf{ZFC}$ + $\mathsf{BPFA}$ + “there are no projective infinite MAD families” is exactly a ${\mathrm{\Sigma }}_1$-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under $\mathsf{BMM}$—there are no ${\mathbf{\Sigma }}_3^1$ infinite MAD families.

中文翻译：

---

可定义的MAD族和强迫公理

我们证明 $\mathsf{零碳燃料}$ + $\mathsf{BPFA}$ （即有界的正确强迫公理）+“没有 ${\mathbf{\Pi }}_2^{\mathrm{1个}}$ 无限的MAD家族”意味着 ${\omega }_{\mathrm{1个}}$是L的一位重要枢机。换句话说，在$\mathsf{BPFA}$ 和反大基数假设有一个 ${\mathbf{\Pi }}_2^{\mathrm{1个}}$无限的MAD家庭。因此，一致性强度$\mathsf{零碳燃料}$ + $\mathsf{BPFA}$ +“没有投影的无限MAD族”恰好是 ${\mathrm{\Sigma }}_{\mathrm{1个}}$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$\mathsf{BMM}$—没有 ${\mathbf{\Sigma }}_3^{\mathrm{1个}}$ 无限的MAD家庭。