SOME SET THEORY WITHOUT MUCH CHOICE (II): A (NOT SO) SHORT–LIVED AXIOM

Generic absoluteness is an appealing source of axioms in set theory. The general motto is: Everything that is possible (forcible) is true. This turns out to be far too strong when the ambient theory is ZFC, so one needs to restrict the class of statements in ‘everything ’ and, if this class of statements is large enough, also the class of forcing extensions under consideration in ‘is forcible’: For example, one can always force CH and ¬CH by very well behaved forcing notions (both can be forced by proper posets, actually), so any form of generic absoluteness as above involving a class of statements to which both CH and ¬CH belong and a class of forcing notions containing, say, all proper posets is false. Now notice that the fact that there is always a partial order forcing CH and a partial order forcing ¬CH uses the Axiom of Choice. It is not at all obvious if the same is true if we just assume ZF in our universe. In fact it is not obvious that, assuming ZF, one can find any example of a sentence for which there are posets P and Q such that P and Q ¬ ... This motivates the following radical form of generic absoluteness, considered b