Gregory trees, the continuum, and Martin's axiom

A Class of Consistent Anti-martin’s Axioms

Both the Continuum Hypothesis and Martin's Axiom allow induc-tive constructions to continue in circumstances where the inductive hypothesis might otherwise fail. The search for useful related axioms procedes naturally in two directions: towards "Super Martin's Axioms ," which extend MA to broader classes of orders; and towards "Anti-Martin's Axioms" (AMA's) which are strictly weaker than CH, bu...

Towers on Trees

We show that (under MA) for any c many dense sets in Laver forcing L there exists a ff-centered Q Ç L such that all the given dense sets are dense in Q . In particular, MA implies that L satisfies MA and does not collapse the continuum and the additivity of the Laver ideal is the continuum. The same is true for Miller forcing and for Mathias forcing. In the case of Miller forcing this involves ...

A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-R1,א1 , which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property R1,א1 . In this paper, we investigate the property R1,א1 . For example, we show that a forcing notion with the property R1,א1 does not add random reals. We prove tha...

Bounded Martin's Maximum, Weak Erd} Os Cardinals, and Ac

We prove that a form of the Erd} os property (consistent with V = LH! 2 ] and strictly weaker than the Weak Chang's Conjecture at !1), together with Bounded Martin's Maximum implies that Woodin's principle AC holds, and therefore 2 @ 0 = @2. We also prove that AC implies that every function f : !1 ! !1 is bounded by some canonical function on a club and use this to produce a model of the Bounde...

Topology Proceedings 9 (1984) pp. 105-121: FEEBLY COMPACT SPACES, MARTIN'S AXIOM, AND "DIAMOND"