The Theory of Countable Analytical Sets

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable ∏_n^1 set of reals, C_n (this is also true for n even, replacing ∏_n^1 by Σ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Δ_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.