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7. Forcing Axioms


    1. Add remark

      Problem 7.05.

      [Mota]
      1. Does Con(ZFC) imply Con(MM for posets of size $\aleph_1$)?

      2. Is there a stationary set preserving poset of size $\aleph_1$ which is not semiproper?
        1. Remark. [Magidor] The analogue of the first question for BPFA has a negative answer, since by work of Goldstern and Shelah BPFA implies the existence of a $\Sigma_2$-reflecting cardinal.
            • Remark. [Krueger] A counterexample for part 2 cannot be constructed in ZFC alone.

                  Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.