Infinity in Mathematics: Is Cantor Necessary? (Conclusion)

Source abstract

Abstract By Godel’s doctrine I mean the view first enunciated in footnote 48a of Godel (1931) that the “true reason” for the incompleteness phenomena is that “the formation of ever higher types can be continued into the transfinite,” both in systems explicitly using types and in systems of set theory such as ZF for which the (cumulative) type structure is implicit in the axioms. For, as Godel says, the “undecidable propositions constructed here become decidable whenever appropriate higher types are added.” Since the undecidable propositions are of finitary character, Godel ‘s doctrine says in effect that the unlimited transfinite iteration of the power-set operation is necessary to account for finitary mathematics.