ABSTRACT

A set is E-closed if it is closed under the schemes of set recursion, i.e. the Kleene schemes S0–S9 (with equality) revised to allow objects of all types. Inadmissible E-closed sets admit: E-closed generic extensions via appropriate forcing notions, solutions to Post's problem via priority arguments, and sidewise E-closed extensions via type-omitting results for logic on E-closed sets. A sample theorem states: suppose L(κ) is countable, E-closed, and satisfies “there is a greatest cardinal and its cofinality exceeds ω”; then there exist δ ≤ κ and T ⊆ δ such that L(κ, T) is the least E-closed set with T as a member.

INTRODUCTION

The schemes of set recursion (or E-recursion) were devised independently by Normann [1] and Moschovakis. When restricted to objects of finite type, they coincide with the Kleene schemes [2] plus equality, the so-called normal case. For each set X, let E(X) be the least set closed under application of the schemes of set recursion and containing X ∪ {x}. E(2ω) is precisely the collection of all sets coded by relations R on 2ω such that R is Kleene recursive in 3E and some real b. The operator E yields a generalization of the Kleene theory to objects of arbitrary type. Let TP (κ) be the set of all sets of objects of type less than κ, and let κE : TP (κ) → {0,1} be 0 on x iff x = ø. Then E(TP (κ)) is in essence the set of all computations needed to define recursion in κE in the sense intended by Kleene.