On Martin's Axiom and Forms of Choice. Issue 3 (13th April 2016)

• Record Type: Journal Article
• Authors: Tachtsis, Eleftherios
• Is Part Of: Mathematical logic quarterly. Volume 62:Issue 3(2016)
• Page Start: 190
• Page End: 203
• Publication Date: 2016-04-13
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.201400115
• Languages: English
• ISSNs: 0942-5616

Abstract

Abstract : Martin's Axiom MA is the statement that for every well‐ordered cardinal κ < 2 ℵ 0, the statement MA ( κ ) holds, where MA ( κ ) is "if ( P, ≤ ) is a c.c.c. quasi order and D is a family of ≤ κ dense sets in P, then there is a D ‐generic filter of P ". In ZFC, the fragment MA ( ℵ 0 ) is provable, but not in general in ZF . In this paper, we investigate the interrelation between MA ( ℵ 0 ) 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 (not necessarily well‐ordered) cardinal n the statement MA n to be "if ( P, ≤ ) is a c.c.c. quasi order with | P | ≤ n, and D is a family of ≤ n dense sets in P, then there is a D ‐generic filter of P ". We then define MA * to be the statement that for every (not necessarily well‐ordered) cardinal n < 2 ℵ 0, we have that MA n holds. We then investigate the set‐theoretic strength of the principle MA * .