Countable ordinals and the analytical hierarchy. I

Alexander S. Kechris

Vol. 60 (1975), No. 1, 223–227

DOI: 10.2140/pjm.1975.60.223

Abstract

The following results are proved, using the axiom of Projective Determinacy: (i) For n ≧ 1, every Π_2n+1¹ set of countable ordinals contains a Δ_2n+1¹ ordinal, (ii) For n ≧ 1, the set of reals Δ_2n¹ in an ordinal is equal to the largest countable Σ_2n¹ set and (iii) Every real is Δ_n¹ inside some transitive model of set theory if and only if n ≧ 4.

Mathematical Subject Classification

Primary: 02K30

Secondary: 02K15, 02K05

Milestones

Received: 29 August 1974

Published: 1 September 1975