On Foreman's maximality principle

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 $κ$, every $κ$-closed forcing of size $2^{<κ}$ collapses some cardinals.

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