Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hamel bases and well-ordering the continuum


Authors: Mariam Beriashvili, Ralf Schindler, Liuzhen Wu and Liang Yu
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 03E15, 03E20, 03E25
DOI: https://doi.org/10.1090/proc/14010
Published electronically: March 9, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In ZF, the existence of a Hamel basis does not yield a well-ordering of $ {\mathbb{R}}$.


References [Enhancements On Off] (What's this?)

  • [1] Andreas Blass, Existence of bases implies the axiom of choice, Axiomatic set theory (Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 31-33. MR 763890
  • [2] Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
  • [3] J. D. Halpern and A. Lévy, The Boolean prime ideal theorem does not imply the axiom of choice., Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 83-134. MR 0284328
  • [4] Horst Herrlich, Axiom of choice, Lecture Notes in Mathematics, vol. 1876, Springer-Verlag, Berlin, 2006. MR 2243715
  • [5] Kenneth Kunen, Random and Cohen reals, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 887-911. MR 776639
  • [6] A. R. D. Mathias, The order extension principle, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1974, pp. 179-183. MR 0360267
  • [7] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
  • [8] David Pincus and Karel Prikry, Lusin sets and well ordering the continuum, Proc. Amer. Math. Soc. 49 (1975), 429-435. MR 0366667
  • [9] Ralf Schindler, Set theory, Universitext, Springer, Cham, 2014. Exploring independence and truth. MR 3243739
  • [10] Schindler, R., Wu, L., and Yu, L., Hamel bases and the principle of dependent choice, preprint, available at
    https://ivv5hpp.uni-muenster.de/u/rds/hamel$ {}_-$basis$ {}_-$2.pdf

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E15, 03E20, 03E25

Retrieve articles in all journals with MSC (2010): 03E15, 03E20, 03E25


Additional Information

Mariam Beriashvili
Affiliation: Ivane Javakhishvili Tbilisi State University and Ilia Vekua Institute of Applied Mathematics, Tbilisi, Georgia

Ralf Schindler
Affiliation: Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

Liuzhen Wu
Affiliation: Institute of Mathematics, Chinese Academy of Sciences, East Zhong Guan Cun Road No. 55, Beijing 100190, People’s Republic of China

Liang Yu
Affiliation: Institute of Mathematical Sciences, Nanjing University, Nanjing, Jiangsu Province 210093, People’s Republic of China

DOI: https://doi.org/10.1090/proc/14010
Received by editor(s): January 17, 2017
Received by editor(s) in revised form: November 3, 2017
Published electronically: March 9, 2018
Additional Notes: The first author gratefully acknowledges support from the Marianne und Dr. Horst Kiesow-Stiftung, Frankfurt a.M
The second author was partially supported by the SFB 878 “Groups, geometry, and actions” from the DFG (Deutsche Forschungsgemeinschaft).
The third author would like to acknowledge the support through the funding Projects NSFC 11321101 and 11401567.
The fourth author gratefully acknowledges support from the National Natural Science Fund of China, No. 11322112 and 11671196 and from a Humboldt Research Fellowship for Experienced Researchers.
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society