Hierarchies of Forcing Axioms, the Continuum Hypothesis and Square Principles
HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES GUNTER FUCHS Abstract. I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete for- cings, in terms of implications and consistency strengths. For the weak hier- archy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson's ordinal reflection principle at !2, and that its effect on the failure of weak squares is very similar to that of Martin's Maximum. 1. Introduction The motivation for this work is the wish to explore forcing axioms for subcom- plete forcings in greater detail. Subcomplete forcing was introduced by Jensen in [18]. It is a class of forcings that do not add reals, preserve stationary subsets of !1, but may change cofinalities to be countable, for example. Most importantly, subcomplete forcing can be iterated, using revised countable support. Examples of subcomplete forcings include all countably closed forcings, Namba forcing (assu- ming CH), Pˇr´ıkr´yforcing (see [19] for these facts), generalized Pˇr´ıkr´yforcing (see [24]), and the Magidor forcing to collapse the cofinality of a measurable cardinal of sufficiently high Mitchell order to !1 (see [10]). Since subcomplete forcings can be iterated, they naturally come with a forcing axiom, the subcomplete forcing axiom, SCFA, formulated in the same way as Mar- tin's axiom or the proper forcing axiom PFA. The overlap between proper forcings and subcomplete forcings is minimal, though. Proper forcings can add reals, which subcomplete forcings cannot.
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