MA(ℵ0) restricted to complete Boolean algebras and choice. Issue 4 (17th November 2021) - Tachtsis, Eleftherios
Abstract: It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment MA ( ℵ 0 ) of Martin's Axiom either in ZFA or in ZF . 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 MA ( ℵ 0 ) restricted to complete Boolean algebras in ZF . Furthermore, we prove that the latter (formally) weaker form of MA ( ℵ 0 ) and the Δ‐system Lemma are independent of each other in ZFA . We also answer open questions from Tachtsis [16] which concern the status of MA ( ℵ 0 ) restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of ZFA and we strengthen some results from the above paper.