Abstract
Assuming the consistency of ZFC we prove the claim in the title by showing the consistency with ZFC of: There exists a set of realsA such that every function fromA toA is order preserving on an uncountable set. We prove related results among which is the consistency with ZFC of: Every function from the reals into the reals is monotonic on an uncountable set.
Similar content being viewed by others
References
U. Avraham, M. Rubin and S. Shelah, in preparation.
J. E. Baumgartner,All ℕ 1 -dense sets of reals can be isomorphic, Fund. Math.79 (1973), 101–106.
F. Galvin and S. Shelah,Some counterexamples in the partition calculus, J. Combinatorial Theory A15 (1973), 167–174.
S. Shelah,Isomorphism of ℕ 1 -dense subsets of reals, Notices Amer. Math. Soc.192 (1979), A-224.
W. Sierpinski,Hypothèse du Continu, Monografie Mathematyczne, Warszawa-Lwow, 1934.
R. M. Solovay and S. Tennenbaum,Iterated Cohen extensions and Souslin’s Problem, Ann. Math.94 (1971), 201–245.
Author information
Authors and Affiliations
Additional information
I would like to thank the United States-Israel Binational Science Foundation for supporting this research by a grant.
Rights and permissions
About this article
Cite this article
Avraham, U., Shelah, S. Martin’s axiom does not imply that every two ℕ1-dense sets of reals are isomorphic. Israel J. Math. 38, 161–176 (1981). https://doi.org/10.1007/BF02761858
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02761858