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.
Similar content being viewed by others
References
Baumgartner, J.E.: Applications of the proper forcing axiom. In: Kunen, K., Vaughn, J.E. (eds.) Handbook of set theoretic topology. Amsterdam: North Holland, 1984
Devlin, K.J.: A Yorkshireman's guide to proper forcing. Proc. 1978 Cambridge Summer School in Set Theory
Goltz, H.J.: Untersuchungen zur Elimination verallgemeinerter Quantoren in Körpertheorien. Wiss. Z. Humboldt-Univ. Math.-Naturwiss. Reihe29, 391–397 (1980)
Kunen, K.: Set theory. Amsterdam: North-Holland, 1980
Luzin, N.: Sur un problème de M. Baire. C. R. Hebdomadaires Séances Acad. Sci. Paris158, 1258–1261, (1914)
Bürger, G.: Thesis, Freiburg 1989
Rapp, A.: The ordered field of real numbers and logics with Malitz-quantifiers. J. Symb. Logic50, 380–389 (1985)
Shelah, S.: Proper forcing. (Lect. Notes Math.940) Berlin, Heidelberg, New York: Springer, 1982
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01622875