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Σ1-separation

Published online by Cambridge University Press:  12 March 2014

Fred G. Abramson*
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

Abstract

Let A be a standard transitive admissible set. Σ1-separation is the principle that whenever X and Y are disjoint Σ1A subsets of A then there is a ⊿1A subset S of A such that XS and YS = ∅.

Theorem. If satisfies Σ-separation, then

(1) If 〈Tn∣n < ω) ϵ A is a sequence of trees on ω each of which has at most finitely many infinite paths in A then the function n ↦ (set of infinite paths in A through Tn) is in A.

(2) If A is not closed under hyperjump and α = OnA then A has in it a nonstandard model of V = L whose ordinal standard part is α.

Theorem. Let α be any countable admissible ordinal greater than ω. Then there is a model of Σ1-separation whose height is α.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1979

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References

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