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Metamathematics of internal theories

  • Vladimir Kanovei
  • Michael Reeken
Chapter
  • 491 Downloads
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

One of the most important metamathematical issues related to any formal theory is the question of consistency: that is, a theory should not imply a contradiction. As long as minimally reasonable set theories are considered, Gödel’s famous incompleteness theorems make it impossible to prove the consistency in any absolute sense, so that usually the results are given in terms of equiconsistency with some other theory, for instance, ZFC. In this Chapter, we prove that the internal theories IST and BST considered above are equiconsistent with ZFC, that is, consistency of ZFC logically implies consistency of both BST and IST. (For the opposite direction, if IST or BST is consistent then obviously so is ZFC as a subtheory of each of IST, BST see Exercise 3.1.3.)

Keywords

Truth Predicate Proper Classis Transitive Model Finite Support Elementary Embedding 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vladimir Kanovei
    • 1
  • Michael Reeken
    • 2
  1. 1.IITP, Institute for Information TransmissionMoscowRussian Federation
  2. 2.Bergische Universität WuppertalFB C MathematikWuppertalGermany

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