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Locally countable models of Σ1-separation

  • Fred G. Abramson (a1) (a2)

Abstract

Let α be any countable admissible ordinal greater than ω. There is a transitive set A such that A is admissible, locally countable, OnA = α, and A satisfies Σ1-separation. In fact, if B is any nonstandard model of KP + ∀xω (the hyperjump of x exists), the ordinal standard part of B is greater than ω, and every standard ordinal in B is countable in B, then HCB ∩ (standard part of B) satisfies Σ-separation.

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References

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[1]Abramson, F. G., Σ 1-separation, this Journal, vol. 44 (1979), pp. 374382.
[2]Barwise, K. J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.
[3]Barwise, J. and Fisher, E., The Shoenfield absoluteness lemma, Israel Journal of Mathematics, vol. 8 (1970), pp. 329339.
[4]Harrington, L. A., An admissible set with no intermediate Σ 1-degrees (in preparation).
[5]Vaught, R., Descriptive set theory in Lω1ω, Cambridge Summer School in Mathematical Logic, Lecture Notes in Mathematics, vol. 337, Springer, Berlin and New York, 1973, pp. 574598.

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