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, but which, when combined with->CH, deny MA. In this paper, we consider restrictions of van Douwen and Fleissner's Undefinable Forcing Axiom which are consistent with the negation of the continuum hypothesis. 1. Introduction. Baumgartner proposed an Anti-Martin's Axiom that he referred to as "The Complete Failure of Martin's Axiom," which asserts that for each c.c.c. order, there is a collection of ω\ dense subsets of that order which cannot be met by any filter. This axiom is clearly an AMA; it follows from CH, is consistent with->CH, and, in conjunction with the negation of CH, implies the failure of MA. Unfortunately, a collection of essentially random subsets of a c.c.c. order yields only a very weak inductive capacity. Another, less well-known, Anti-Martin's Axiom was introduced by van Douwen and Fleissner. Analyzing a model introduced by Bell and Kunen, they extracted an axiom they referred to as "The Definable Forcing Axiom," which captures some of the properties of that model. In this paper, we examine several other axioms closely related to DFA and capturing other facets of this model.