We prove that a form of the Erdos property (consistent with V = L[H] and strictly weaker than the Weak Chang's Conjecture at ?), together with Bounded Martin's Maximum implies that Woodin's principle ? holds, and therefore 2 = ?. We also prove that ? implies that every function f: ? ? ? is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.
- Bounded Martin's Maximum
- Bounding by canonical functions
- Erdos cardinals