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.
@article{bwmeta1.element.bwnjournal-article-fmv159i3p231bwm,
author = {R. Berr and Fran\c coise Delon and J. Schmid},
title = {Ordered fields and the ultrafilter theorem},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {231-241},
zbl = {0940.12004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv159i3p231bwm}
}
Berr, R.; Delon, Françoise; Schmid, J. Ordered fields and the ultrafilter theorem. Fundamenta Mathematicae, Tome 159 (1999) pp. 231-241. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv159i3p231bwm/
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