Feferman on the Indefiniteness of CH

Feferman is arguably the foremost critic of set theory since Weyl. His paper “Is the continuum hypothesis a definite mathematical problem?” provides a useful overview of his negatie answer, drawing on many papers over many years. The main thesis is that the continuum hypothesis (CH) is an “indefinite” statement and that the underlying reason is that the “concept of arbitrary set [which is essential to the formulation of CH] is vague or underdetermined” and “there is no way to sharpen it without violating what it is supposed to be about.” (1) In what follows I am will exposit and extend Feferman’s critique, argue that each component fails, and conclude that when the dust settles the entire case rests on the claim that the concept of natural number is clear while the concept of arbitrary sets of natural numbers is not clear. My stance on this final resting point is captured in the above quotation from Markov.