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Σ1-separation

Published online by Cambridge University Press:  12 March 2014

Fred G. Abramson
Fred G. Abramson*
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
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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 X ⊆ S and Y ∩ S = ∅.

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 = OnAthen 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.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 3 , September 1979, pp. 374 - 382
Copyright
Copyright © Association for Symbolic Logic 1979

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References

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