Some Set Theories Are More Equal

The proof of independence of the Continuum Hypothesis by Paul Cohen in 1963 and the avalanche of additional independence results was the trigger to a very intensive discourse about the future of Set Theory and its foundational role in Mathematics. It will be only fair to say that while the invention of forcing initiated the blooming of Set Theory as a technical mathematical discipline, it had a shocking effect on the perception of Set Theory as the framework in which all of Mathematics (or at least a major part of it) can be formalized.The prevailing tone of these reactions is that Set Theory (at least as formalized by the Zermelo-Fraenkel axioms, denoted by ZFC) lost its uniqueness.What made the shock even more dramatic was the fact the proof of the independence of the Continuum Hypothesis showed that ZFC gives us very little information about the size of the Continuum and as far as the axioms are concerned there is a huge range of possibilities like 2א0 = א1, 2א0 = א2 . . . 2א0 = א211086 . . . 2א0 = אω+1720 = אω1 . Here are two quotes from many: Such results show that axiomatic Set Theory is hopelessly incomplete. . . If there are a multitude of set theories then none of them can claim the central place in Mathematics. (Mostowski-1967 [29]) Beyond classical analysis there is an infinity of different mathematics and for the time being no definitive reason compels us to chose one rather than another. (Dieudonné-1976 [?]) This shock is somewhat puzzling because the fact that ZFC is not complete and therefore contains some independent statements was well known in view of Gödel’s incompleteness theorems. Gödel himself in the early version of [15] is confident that CH is independent of ZFC.