Abstract

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω 1 with ω 1 generators, then there exists an uncountable $X \subseteq \omega_1$ , such that either [ X] ω ∩ I = ⊘ or $\lbrack X\rbrack^\omega \subseteq I$