Languages, Structures, and Models

  • Gaisi Takeuti
  • Wilson M. Zaring
Part of the Graduate Texts in Mathematics book series (GTM, volume 1)

Abstract

In Sections 2–15 we developed Zermelo-Fraenkel set theory in a very simple language and we proved Gödel’s consistency results. In Sections 16–18 we enriched the original language by adding individual constants. In this enriched language we developed a rather powerful technique, Cohen’s forcing, with which we proved one of Cohen’s independence results. In this section we will introduce a language that is richer than any we have used heretofore. We will briefly outline the development of Zermelo-Fraenkel theory in this language and define certain concepts for later use. One of our objectives is to explain an earlier claim that the theories ZF {α} are slightly different formulations of Zermelo-Fraenkel set theory than that of Sections 2–15. Our main objective, however, is the introduction of a language 𝓛 that is of great value in the development of a very general method for attacking certain problems in set theory.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • Gaisi Takeuti
    • 1
  • Wilson M. Zaring
    • 1
  1. 1.University of IllinoisUSA

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