Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T02:17:51.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Independently axiomatizable ℒω1,ω theories

Published online by Cambridge University Press:  12 March 2014

Greg Hjorth  and
Ioannis A. Souldatos
Greg Hjorth
Affiliation:
Department of Mathematics, The University of Melbourne, Parkville, Vic 3010, Australia, E-mail: [email protected]
Ioannis A. Souldatos
Affiliation:
273 Wissink Hall, Office 261 Mathematics Department, Minnesota State UniversityMankato, Mn 56001, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In partial answer to a question posed by Arnie Miller [4] and X. Caicedo [2] we obtain sufficient conditions for an ℒω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every ℒω1,ω theory in a countable language has an independent axiomatization. The second, this time outright in ZFC, every intersection of a family of Borel sets can be formed as the intersection of a family of independent Borel sets.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1273 - 1286
Copyright
Copyright © Association for Symbolic Logic 2009

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]Becker, H. and Kechirs, A., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, 1996.CrossRefGoogle Scholar
[2]Caicedo, X., Independent sets of axioms in Lk∝, Canadian Mathematical Bulletin, vol. 24 (1981), no. 2, pp. 219–223.CrossRefGoogle Scholar
[3]Kechris, Alexander S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[4]Miller, Arnold W., http://www.math.wise.edu/~miller/res/problem.pdf, This webpage contains a list of interesting problems in Set Theory and Model Theory.Google Scholar
[5]Reznikoff, M. I., Tout ensemble de formules de la logique classique est equivalent à un ensemble independant, Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 260 (1965), pp. 2385–2388.Google Scholar