Juhász’s topological generalization of Neumer’s theorem may fail in $\mathsf {ZF}$
HTML articles powered by AMS MathViewer
- by Eleftherios Tachtsis PDF
- Proc. Amer. Math. Soc. 148 (2020), 1295-1310 Request permission
Abstract:
In set theory without the Axiom of Choice ($\mathsf {AC}$), we investigate the open problem of the deductive strength of Juhász’s topological generalization of Neumer’s Theorem from his paper On Neumer’s Theorem [Proc. Amer. Math. Soc. 54 (1976), 453–454].
Among other results, we show that Juhász’s Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of $T_{1}$ spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen’s Countable Choice Principle, but does not imply either the full van Douwen’s Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász’s Theorem (for $T_{1}$ spaces) implies each of the following principles: “$\aleph _{1}$ is a regular cardinal”, “every infinite set is weakly Dedekind-infinite”, and “every infinite linearly ordered set is Dedekind-infinite”. We also establish that Juhász’s Theorem for $T_{2}$ spaces is not provable in $\mathsf {ZF}$.
In contrast to the above results, we show that Neumer’s Theorem and Juhász’s Theorem for compact $T_{1}$ spaces are both provable in $\mathsf {ZF}$.
References
- G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. (Szeged) 17 (1956), 139–142 (German). MR 82450
- Horst Herrlich, Axiom of choice, Lecture Notes in Mathematics, vol. 1876, Springer-Verlag, Berlin, 2006. MR 2243715
- Paul Howard, Kyriakos Keremedis, Jean E. Rubin, Adrienne Stanley, and Eleftherios Tatchtsis, Non-constructive properties of the real numbers, MLQ Math. Log. Q. 47 (2001), no. 3, 423–431. MR 1847458, DOI 10.1002/1521-3870(200108)47:3<423::AID-MALQ423>3.0.CO;2-0
- Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. With 1 IBM-PC floppy disk (3.5 inch; WD). MR 1637107, DOI 10.1090/surv/059
- Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0396271
- I. Juhász, On Neumer’s theorem, Proc. Amer. Math. Soc. 54 (1976), 453–454. MR 394569, DOI 10.1090/S0002-9939-1976-0394569-X
- I. Juhász and A. Szymanski, The topological version of Fodor’s theorem, More sets, graphs and numbers, Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin, 2006, pp. 157–174. MR 2223392, DOI 10.1007/978-3-540-32439-3_{8}
- A. Karagila, Fodor’s lemma can fail everywhere, Acta Math. Hungar. 154 (2018), no. 1, 231–242. MR 3746534, DOI 10.1007/s10474-017-0768-5
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- A. Lévy, Axioms of multiple choice, Fund. Math. 50 (1961/62), 475–483. MR 139528, DOI 10.4064/fm-50-5-475-483
- Walter Neumer, Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Math. Z. 54 (1951), 254–261 (German). MR 43860, DOI 10.1007/BF01574826
- David Pincus, Adding dependent choice, Ann. Math. Logic 11 (1977), no. 1, 105–145. MR 453529, DOI 10.1016/0003-4843(77)90011-0
- Eleftherios Tachtsis, A note on the deductive strength of the Juhász–Szymanski topological version of Fodor’s theorem, preprint.
Additional Information
- Eleftherios Tachtsis
- Affiliation: Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
- MR Author ID: 657401
- Email: ltah@aegean.gr
- Received by editor(s): February 14, 2019
- Received by editor(s) in revised form: May 22, 2019, July 14, 2019, and July 21, 2019
- Published electronically: October 28, 2019
- Communicated by: Heike Mildenberger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1295-1310
- MSC (2010): Primary 03E25; Secondary 03E35, 54A35
- DOI: https://doi.org/10.1090/proc/14794
- MathSciNet review: 4055956