It is easy to extend to
arbitrary structures A = ⟨A,R1,⋯,Rlf1,⋯,fm⟩ 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 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.
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