The Suslin-Kleene theorem for V _κ with cofinality (κ) = ω

Chen Chung Chang and Yiannis (John) Nicolas Moschovakis

Vol. 35 (1970), No. 3, 565–569

DOI: 10.2140/pjm.1970.35.565

Abstract

It is easy to extend to arbitrary structures A = ⟨A,R₁,⋯,R_lf₁,⋯,f_m⟩ the concepts of ∏ ₁¹ and inductively definable relations, which are familiar for the structure of the integers. The second author showed in a recent paper that these two concepts coincide for countable A that satisfy certain mild definability conditions—this is a generalization of the classical Suslin-Kleene theorem. Here we generalize the Suslin-Kleene theorem in a different direction.

Main Result. Let V _κ be the set of sets of rank less than κ, i.e., V ₀ = ϕ,V _ξ+1 = power of V _ξ,V _κ = ⋃ _ξ<κV _ξ, if κ is limit. The classes of inductively definable and ∏ ₁¹ relations on the structure 𝒱_κ = ⟨V _κ,∈↑ V _κ⟩(κ ≥ ω) coincide if and only if κ is a limit ordinal with cofinality ω.

This implies several corollaries about the class of ∏ ₁¹ relations on V _κ, when cofinality (κ) = ω, e.g., that it has the reduction property.

Mathematical Subject Classification

Primary: 02.77

Milestones

Received: 26 September 1969

Revised: 1 May 1970

Published: 1 December 1970