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

We present several models which satisfy CH and some ♦-like principles while others fail, answering a question of Moore, Hrušák and Džamonja.

It is proved, under Martin's Axiom, that all gaps in are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin's Axiom the Baire categoricity of all Δ31 non-Δ31-complete sets of reals requires a weakly compact cardinal.

We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”.

We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...) example of a so-called boldface bounded forcing axiom implying 20=2. (shrink)