Σ1-separation

Journal of Symbolic Logic 44 (3):374 - 382 (1979)

Abstract
Let A be a standard transitive admissible set. Σ 1 -separation is the principle that whenever X and Y are disjoint Σ A 1 subsets of A then there is a Δ A 1 subset S of A such that $X \subseteq S$ and $Y \cap S = \varnothing$ . Theorem. If A satisfies Σ 1 -separation, then (1) If $\langle T_n\mid n is a sequence of trees on ω each of which has at most finitely many infinite paths in A then the function $n\mapsto$ (set of infinite paths in A through T n ) is in A. (2) If A is not closed under hyperjump and α = On A 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 Σ 1 -separation whose height is α.
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DOI 10.2307/2273130
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The Problem of Predicativity.Joseph R. Shoenfield - 1961 - In Bar-Hillel, Yehoshua & [From Old Catalog] (eds.), Essays on the Foundations of Mathematics. Jerusalem, Magnes Press, Hebrew University;. pp. 132--139.
Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.

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