Projective Forcing

We study the projective posets and their properties as forcing notions. We also define Martin’s axiom restricted to projective sets, MA(proj), and show that this axiom is weaker than full Martin’s axiom by proving the consistency of ZFC + XH + MA(proj) with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists K < 2No such that 2No < 2K”.

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