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

摘要

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