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The Arithmetization of Model Theory

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

Abstract

In Section 12 we developed a theory of internal models and gave meaning to the idea of a class M being a standard transitive model of ZF. For 𝓜 to be a standard model of ZF means in particular that 𝓜 is a model of Axiom 5, the Axiom Schema of Replacement. Since Axiom 5 is a schema “M is a standard model of ZF” is a meta-statement asserting that a certain infinite collection of sentences of ZF hold. Can this metastatement be formalized in ZF, that is, can “𝓜 is a standard model of ZF” be expressed as a single sentence in ZF? This question is as yet unresolved. It can be resolved however if 𝓜 is a set, that is, “m is a standard model of ZF” can be expressed as a single sentence in ZF. The basic objective of this section is to produce such a sentence. Our approach is to assign Gödel numbers to the well formed formulas of our language. This assignment will be made by the mapping J of Definition 15.2.

Keywords

Induction Hypothesis Internal Model Formation Rule Axiom Schema Intuitive Notion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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