  Using the method of inner models Gödel showed that ¬PM holds in L.  It follows that ZFC+ ¬PM is mutually interpretable with ZFC.  But what about PM?  To show that ZFC+PM is mutually interpretable with ZFC a natural approach would be to follow the approach used for CH and build an outer model of ZFC that satisfies PM.  However, it is known that this cannot be done starting with ZFC alone.  For it turns out (by a result of Shelah (1984)) that ZFC+PM implies the consistency of ZFC and this implies, by the second incompleteness theorem, that ZFC+PM is not interpretable in ZFC.  In a sense we have here a case of the independence of independence.  More precisely, even if we assume that ZFC is consistent we cannot (in contrast to the case of CH) prove that PM is independent of ZFC.  To establish the independence of PM from ZFC we need to assume the consistency of a stronger theory, namely, that of   ZFC + “There is a strongly inaccessible cardinal”.  For it turns out that ZFC+PM lies not in the interpretability degree of ZFC but rather in that of ZFC + “There is a strongly inaccessible cardinal”.  To summarize: While CH is a case of the second type independence, PM is a case of the first type independence; it is similar to Con(ZFC) in that it is a sentence φ such that only one of φ or ¬φ leads to a jump in strength, only now there are two differences; the jump lands in a degree that is much stronger and it is represented by a natural theory.