my forty years on his shoulders - Department of Mathematics

37P.J. Cohen proved that if ZF is consistent then so is ZF + ¬AxC and ZFC +¬CH, thus complementing Gödel’s results. See (Cohen 1963-1964). The proofdoes not readily give an interpretation of ZF + ¬AxC, or of ZFC + ¬CH in ZF. Itcan be converted into such an interpretation by a general method whereby undercertain conditions (met here), if the consistency of every given finite subsystem ofone system is provable in another, then the first system is interpretable in theother (see (Feferman 1960)).Again, the question arises as to how simple can an interpretation be of ZF+ ¬AxC or of ZFC + ¬CH, in ZF, with abbreviations allowed in the presentationof the interpretation? Again this is far from clear. And how does this questioncompare with the previous question?There is another kind of complexity issue associated with the CH that is ofinterest. First some background. It is known that every 3 quantifier sentence inprimitive notation ∈,=, is decided in a weak fragment of ZF. See (Gogol 1979),(Friedman 2003a). Also there is a 5 quantifier sentence in ∈,= that is not decidedin ZFC (it is equivalent to the existence of a subtle cardinal over ZFC). See(Friedman 2003b). It is also known that AxC can be written with five quantifiersin ∈,=, over ZFC. See (Maes 2007).The question is: how many quantifiers are needed to express CH overZFC, in ∈,=? We can also ask this and related questions where abbreviations areallowed.Most mathematicians instinctively take the view that since CH is neitherprovable nor refutable from the standard axioms for mathematics (ZFC), theultimate status of CH has been settled and there is nothing left to ponder.