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TheL <ω-theory of the class of Archimedian real closed fields

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Abstract

For the classA of uncountable Archimedian real closed fields we show that the statement “TheL <ω-theory ofA is complete” is independent of ZFC. In particular we have the following results:

Assuming the Continuum-Hypothesis (CH)

is incomplete. Conversely it is possible to build a model of set theory in which

is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case

turns out to be equivalent to the elementary theory of the real numbers ℝ (by a quantifier-elimination procedure).

Formally:

is incomplete.

is complete and decidable.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Freiburg, Albertstrasse 23b, D-7800, Freiburg, Federal Republic of Germany

    Gerd Bürger

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  1. Gerd Bürger
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Bürger, G. TheL <ω-theory of the class of Archimedian real closed fields. Arch Math Logic 28, 155–166 (1989). https://doi.org/10.1007/BF01622875

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  • DOI: https://doi.org/10.1007/BF01622875

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