It is shown that in ZF Martin's $ \aleph_{0}^{}$-axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2X of a 2-point discrete space are Baire; and that the latter property implies the following: the axiom of countable choice for finite sets, power sets of infinite sets are Dedekind-infinite, there are no amorphous sets, and weak forms of the Kinna-Wagner principle.

It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in. In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in. Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in.We also answer open questions (...) from Tachtsis [16] which concern the status of restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of and we strengthen some results from the above paper. (shrink)

Martin's Axiom math formula is the statement that for every well-ordered cardinal math formula, the statement math formula holds, where math formula is “if math formula is a c.c.c. quasi order and math formula is a family of math formula dense sets in P, then there is a math formula-generic filter of P”. In math formula, the fragment math formula is provable, but not in general in math formula. In this paper, we investigate the interrelation between math formula and various (...) choice principles. In the choiceless context, it makes sense to drop the requirement that the cardinal κ be well-ordered, and we can define for any cardinal math formula the statement math formula to be “if math formula is a c.c.c. quasi order with math formula, and math formula is a family of math formula dense sets in P, then there is a math formula-generic filter of P”. We then define math formula to be the statement that for every cardinal math formula, we have that math formula holds. We then investigate the set-theoretic strength of the principle math formula. (shrink)