SELECTIVELY (a)-SPACES FROM ALMOST DISJOINT FAMILIES ARE NECESSARILY COUNTABLE UNDER A CERTAIN PARAMETRIZED WEAK DIAMOND PRINCIPLE

摘要

The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that 2(aleph 0) < 2(aleph 1) alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of this specific weak diamond principle (which is consistent with the negation of the Continuum Hypothesis, apart from other features).