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Archive for Mathematical Logic

, Volume 49, Issue 3, pp 283–289 | Cite as

On the axiom of union

  • Greg Oman
Article
  • 89 Downloads

Abstract

In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:= ZFC minus union. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.

Keywords

Axiom of union Transitive closure Inaccessible cardinal 

Mathematics Subject Classification (2000)

Primary: 03E30 Secondary: 03E35 

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References

  1. 1.
    Barwise J.: Admissible Sets and Structures. An Approach to Definability Theory, Perspectives in Mathematical Logic. Springer, Berlin (1975)Google Scholar
  2. 2.
    Enderton H.: Elements of Set Theory. Academic Press, New York (1977)zbMATHGoogle Scholar
  3. 3.
    Felgner, U.: Models of ZF—Set Theory, Springer Lecture Notes in Mathematics 223 (1971)Google Scholar
  4. 4.
    Gonzalez C.G.: The union axiom in Zermelo set theory. Z. Math. Logik Grundlag. Math. 36(4), 281–284 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Jech, T.: Set Theory—Third Millennium Edition. Springer Monographs in Mathematics, New York (2002)Google Scholar
  6. 6.
    Kunen K.: Set Theory. An Introduction to Independence Proofs, Studies in Logic and the Foundation of Mathematics. North-Holland, Amsterdam (1980)Google Scholar
  7. 7.
    Shoenfield J.: Mathematical Logic. Addison-Wesley, Reading (1967)zbMATHGoogle Scholar
  8. 8.
    Woodin W.H.: The continuum hypothesis I. Not. Am. Math. Soc. 48(6), 567–576 (2001)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensUSA

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