On Foreman's maximality principle

Abstract

In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is consistent that every $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $\kappa$, every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinals.