Martin's axiom and almost disjoint families

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...

Analytic and coanalytic families of almost disjoint functions

If F ⊆ N N is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ N N, no member of which is covered by finitely many functions from F , there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f (k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventua...

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"

Topology Proceedings 9 (1984) pp. 297-306: TWO NORMAL LOCALLY COMPACT SPACES UNDER MARTIN'S AXIOM