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    Larson, Paul B. 2012. Sets and Extensions in the Twentieth Century.

    Addison, J.W. 2004. Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. Annals of Pure and Applied Logic, Vol. 126, Issue. 1-3, p. 77.

    Duparc, J. 2003. A hierarchy of deterministic context-free ω-languages. Theoretical Computer Science, Vol. 290, Issue. 3, p. 1253.

    Kechris, Alexander S. Solovay, Robert M. and Steel, John R. 1981. Cabal Seminar 77 – 79.

    1980. Descriptive Set Theory.


Separation principles and the axiom of determinateness1

  • Robert A. van Wesep (a1) (a2)
  • DOI:
  • Published online: 01 March 2014

Let Γ be a class of subsets of Baire space (ωω) closed under inverse images by continuous functions. We say such a Γ is continuously closed. Let , the class dual to Γ, consist of the complements relative to ωω of members of Γ. If Γ is not selfdual, i.e., , then let . A continuously closed nonselfdual class Γ of subsets of ωΓ is said to have the first separation property [2] if

The set C is said to separate A and B. The class Γ is said to have the second separation property [3] if

We shall assume the axiom of determinateness and show that if Γ is a continuously closed class of subsets of ωω and then

(1) Γ has the first separation property iff does not have the second separation property, and

(2) either Γ or has the second separation property.

Of course, (1) and (2) taken together imply that Γ and cannot both have the first separation property.

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This paper represents research done at the University of California, Berkeley, under the guidance of Professor J. W. Addison. It is part of the author's Ph.D. thesis. The author wishes to thank Professor Addison for his guidance and encouragement.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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