Ordered fields and the ultrafilter theorem

R. Berr; Françoise Delon; J. Schmid

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 3, page 231-241
  • ISSN: 0016-2736

Abstract

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We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.

How to cite

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Berr, R., Delon, Françoise, and Schmid, J.. "Ordered fields and the ultrafilter theorem." Fundamenta Mathematicae 159.3 (1999): 231-241. <http://eudml.org/doc/212331>.

@article{Berr1999,
abstract = {We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.},
author = {Berr, R., Delon, Françoise, Schmid, J.},
journal = {Fundamenta Mathematicae},
keywords = {axiom of choice; formally real ring; semi-real; ordering; total preodering; ultrafilter theorem; JFM 52.0120.05; Artin-Schreier theorem},
language = {eng},
number = {3},
pages = {231-241},
title = {Ordered fields and the ultrafilter theorem},
url = {http://eudml.org/doc/212331},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Berr, R.
AU - Delon, Françoise
AU - Schmid, J.
TI - Ordered fields and the ultrafilter theorem
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 3
SP - 231
EP - 241
AB - We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.
LA - eng
KW - axiom of choice; formally real ring; semi-real; ordering; total preodering; ultrafilter theorem; JFM 52.0120.05; Artin-Schreier theorem
UR - http://eudml.org/doc/212331
ER -

References

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  1. [1] E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg 5 (1927), 100-115. Zbl52.0122.01
  2. [2] E. Artin, Schreier, O. Algebraische Konstruktion reeller Körper, ibid., 85-99. 
  3. [3] T. Jech, The Axiom of Choice, North-Holland, 1973. Zbl0259.02051
  4. [4] H. Lombardi and M.-F. Roy, Constructive elementary theory of ordered fields, in: Effective Methods in Algebraic Geometry, Progr. Math. 94, Birkhäuser, 1991, 249-262. 
  5. [5] T. Sander, Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma, J. Pure Appl. Algebra 73 (1991), 165-180. Zbl0761.12004
  6. [6] A. Tarski, Prime ideal theorems for set algebras and ordering principles, preliminary report, Bull. Amer. Math. Soc. 60 (1954), 391. 

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