Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T17:22:21.953Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abramson, F. G., Sacks forcing does not always produce a minimal upper bound, Advances in Mathematics (to appear).Google Scholar
[2]Abramson, F. G., Locally countable models of Σ1-separation (to appear).Google Scholar
[3]Friedman, H., Countable models of set theories, Cambridge Summer School in Mathematical Logic, 1971, Lecture Notes in Mathematics, no. 337 Springer-Verlag, Berlin and New York, 1973, pp. 539–573.Google Scholar
[4]Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113–129.Google Scholar
[5]Harrington, L. A., An admissible set with no intermediate Σ1-degrees (in preparation).Google Scholar
[6]Jensen, R. B. and Solvay, R. M., Some applications of almost disjoint sets, Mathematical logic and foundation of set theory, Proceedings of an International Colloquium, Jerusalem (Bar-Hillel, , Editor), North-Holland, Amsterdam, 1968, pp. 84–104.Google Scholar
[7]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[8]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[9]Shoenfield, J. R., The problem of predicativity, Essays on the foundations of mathematics (Bar-Hillel, , Poznanski, , Rabin, , and Robinson, , Editors), Magnes Press, Jerusalem (distributed by North-Holland, Amsterdam), 1961, pp. 132–139.Google Scholar
[10]Simpson, S. G., Notes on subsystems of analysis, mimeographed, 1973.Google Scholar
[11]Barwise, K. J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar