Cardinalities in the projective hierarchy

Abstract

We show that the “effective cardinality” of the collection of sets is strictly bigger than the effective cardinality of the . The phrase effective cardinality is vague but can be made exact in the usual ways. For instance:Theorem 1.1. Assume ADL(ℝ)Then in L(ℝ) there is no injection.A few years ago Tony Martin showed a similar result, establishing the non-existence of an injection from to for m sufficiently larger than n. His method did not seem to work for m = n + 1.This present paper gives level by level calculations for the projective hierarchy, but it too falls short of a complete analysis, in as much as it leaves the position of the effective cardinals in the Wadge degrees largely obscure. At the low levels it takes some time for any new cardinals to appear. Whenever Γ1, Γ2 are non-trivial Wadge degrees strictly included in one has.