On the theory of ∏_3^1 sets of reals

Abstract

Assuming that ∀x Є ω^ω (x^# exists), let u_ɑ be the ɑth uniform indiscernible (see [3] or [2] ). A canonical coding system for ordinals < u_ω can be defined by letting W0_ω = {w Є ω^ω: w = (n, x^#), for some n Є ω, x Є ω^ω} and for w = (n, x^#) є W0_ω, │w│ = Ƭ^L_n [x](u_l',... , u_k_n), where T_n is the nth term in a recursive enumeration of all terms in the language of ZF + V = L [x], x a constant, taking always ordinal values. Call a relation P(ξ x), where ~varies over u^ω and x over ω^ω, ∏^1_k if P^*(w, x)⇔ w Є W0_ω Λ P(│w│, x) is ∏^1_k. An ordinal ξ < u_ω is called Δ^1_k if it has a Δ^1_k notation i.e. ∃ w Є W0_ω (w Є Δ^1_k Λ │w│ = ξ).