A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees

作者：Yorioka Teruyuki^*

来源：Annals of Pure and Applied Logic, 2010, 161(4): 469-487.

DOI：10.1016/j.apal.2009.02.006

• Martin's Axiom and its fragments (omega(1), omega(1))-gaps
• Unbounded families
• Entangled sets of reals
• Adding no random reals
• Non-special Aronszajn trees

We introduce a property of forcing notions, called the anti-R(1,aleph 1), which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property R(1,aleph 1).
In this paper, we investigate the property R(1,aleph 1). For example, we show that a forcing notion with the property R(1,aleph 1) does not add random reals. We prove that it is consistent that every forcing notion with the property R(1,aleph 1) has precaliber aleph(1) and MA aleph(1), for forcing notions with the property R(1,aleph 1) fails. This negatively answers a part of one of the classical problems about implications between fragments of MA aleph(1).

• 出版日期2010-1