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First-Order Continuous Induction and a Logical Study of Real Closed Fields

  • Saeed SalehiEmail author
  • Mohammadsaleh Zarza
Original Paper

Abstract

Over the last century, the principle of “continuous induction” has been studied by different authors in different formats. All of these different forms are equivalent to one of the three versions that we isolate in this paper. We show that one of the three forms of continuous induction is weaker than the other two by proving that it is equivalent to the Archimedean property, while the other two stronger versions are equivalent to the completeness property (the supremum principle) of the real numbers. We study some equivalent axiomatizations for the first-order theory of real closed fields and show that some first-order formalization of continuous induction is able to completely axiomatize it (over the theory of ordered fields).

Keywords

First-order logic Complete theories Axiomatizing the field of real numbers Continuous induction Real closed fields 

Mathematics Subject Classification

03B25 03C35 03C10 12L05 

Notes

Acknowledgements

The authors are very much grateful to the anonymous referee of the Bulletin for carefully reading the article and for many helpful suggestions which improved the results and made the paper much more readable.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Research Institute for Fundamental SciencesUniversity of TabrizTabrizIran
  2. 2.Department of MathematicsUniversity of TabrizTabrizIran

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