Vol. 35, No. 3, 1970

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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

It is easy to extend to arbitrary structures A = A,R1,,Rlf1,,fmthe 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 𝒱κ = 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
Received: 26 September 1969
Revised: 1 May 1970
Published: 1 December 1970
Chen Chung Chang
Yiannis (John) Nicolas Moschovakis