A generalization of Martin’s Axiom

Abstract

Abstract. We define the ℵ 1.5 -chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom; in fact, MA 1.5 <κ implies MA <κ . Also, MA 1.5 <κ implies certain uniform failures of club-guessing on ω 1 that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal κ ≥ ω 2 such that µ ℵ0 < κ for all µ < κ and such that ♦({α < κ : cf(α) ≥ ω 1 }) holds, that there is a proper ℵ 2 -c.c. partial order of size κ forcing 2 ℵ0 = κ together with MA 1.5 <κ .1. A generalization of Martin's Axiom. And some of its applications.Martin's Axiom, often denoted by MA, is the following very wellknown and very classical forcing axiom: If P is a partial order (poset, for short) with the countable chain condition 1 and D is a collection of size less than 2 ℵ 0 consisting of dense subsets of P, then there is a filterMartin's Axiom is obviously a weakening of the Continuum Hypothesis. Given a cardinal λ, MA λ is obtained from considering, in the above formulation of MA, collections D of size at most λ rather than of size less than 2 ℵ 0 . Martin's Axiom becomes interesting when 2 ℵ 0 > ℵ 1 . MA ℵ 1 was the first forcing axiom ever considered ([9]). As observed by D. Martin, the consistency of MA together with 2 ℵ 0 > ℵ 1 follows from generalizing the Solovay-Tennenbaum construction of a model of Suslin's Hypothesis by iterated forcing using finite supports ([14]). Since then, a plethora of applications of MA (+ 2 ℵ 0 > ℵ 1 ) have been

Adding many Baumgartner clubs

“…It is worth pointing out that the stronger forcing axiom MA 1.5 λ implies even stronger failures of club guessing ( [3]). …”

“…Add B (X ) can be seen as a forcing adding Baumgartner clubs, indexed by ordinals in X , by using finite supports and with countable subsets of X as side conditions. The definition of Add B (X ) is a streamlined version of the constructions from [2] or [3]. In fact, it is simple enough that it is an actual product (Lemma 3.5), whereas the constructions in [2] and [3] certainly are not.…”

“…The method of finite-support iterated forcing with symmetric systems of structures as side conditions (see [2] and [3]) can be used to answer this question. This method enables one to force without collapsing cardinals in such a way that the resulting model satisfies FA(P) λ , for an arbitrarily fixed λ, for various well-behaved proper partial orders P with the ℵ 2 -c.c.…”

“…This method enables one to force without collapsing cardinals in such a way that the resulting model satisfies FA(P) λ , for an arbitrarily fixed λ, for various well-behaved proper partial orders P with the ℵ 2 -c.c. Background information on this method can be found in [2] and [3], so I will not say anything about it here. It turns out that B is such a well-behaved forcing.…”

“…It turns out that B is such a well-behaved forcing. More specifically, there is a class of proper posets, which in [3] we refer to as the posets having the ℵ 1.5 -c.c., to which B belongs, and such that the forcing axiom MA({P | P has the ℵ 1.5 -c.c.}) λ -which in [3] we call MA 1.5 λ -is consistent for arbitrarily chosen λ (cf.…”