Projective forcing
The authors study Martin's axiom (MA) restricted to all ccc partial orders which have a projective definition, denoted MA(proj). It is shown that, starting from a model of ZFC, MA(proj) can be forced by a finite-support iteration of ccc projective forcings. It is also shown that consistently MA(proj) is strictly weaker than MA. Several arguments for this are given, some of them, but not all, involving the existence of a weakly compact cardinal. E.g. the consistency of MA(proj) with the negation of CH and the existence of a Suslin tree is proved, assuming Con(ZF+ \(\exists \kappa\) weakly compact).
scientific article; zbMATH DE number 1054203
scientific article