A coding theorem for measures

Abstract

Assuming ZF + DC + AD Moschovakis (see [Ml]) has shown that if there is a surjection π : R → λ from the reals (R = ω^ω in this paper) onto an ordinal λ, then there is a surjection π^* : R → p(λ) from the reals onto the power set of λ. Let us denote by β(λ) the set of ultrafilters on λ. The question was raised whether there is an analog of Moschovakis' Theorem for β(λ), i.e. if there is a surjection from R onto λ, is there one from R onto β(λ)? Martin showed that this cannot be proved in ZF + DC + AD alone because if V = L(R) and λ= ol, there is no surjection of R onto β(λ).

Additional details