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|Paper IPM / M / 16082||
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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