Abstract
In ZF (i.e. the Zermelo–Fraenkel set theory without the Axiom of Choice (AC)), we investigate the set-theoretic strength of a generalized version of Hindman's theorem and of certain weaker forms of this theorem, which were introduced by Fernández-Bretón [8], with respect to their interrelation with several weak choice principles. In this direction, we determine the status of (this general version of) Hindman's theorem (and of weaker forms) in certain permutation models of \(\mathbf{ZFA} + \neg\mathbf{AC}\) and transfer the results to ZF, strengthen some results of [8] and settle a related open problem from Howard and Rubin [10]; thus filling the gap in information in both [8] and [10].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Baumgartner, A Short Proof of Hindman’s Theorem, J. Combin. Theory Ser. A, 17 (1974), 384–386.
A. R. Blass, J. L. Hirst and S. G. Simpson, Logical analysis of some theorems of combinatorics and topological dynamics, Contemp. Math., 65 (1987), 125–156.
J. Brot, M. Cao and D. Fernández-Bretón, Finiteness classes arising from Ramseytheoretic statements in set theory without choice, Ann. Pure Appl. Logic, 172 (2021), 102961.
W. W. Comfort, Some recent applications of ultrafilters to topology, in: General Topology and its Relations to Modern Analysis and Algebra IV, Proceedings of the Fourth Prague Topological Symposium (1976), Part A: Invited Papers (J. Novák, editor), Lecture Notes in Mathematics Vol. 609, Springer (Berlin, 1977), pp. 34-42.
S. G. Da Silva, A topological statement and its relation to certain weak choice principles, Questions Answers Gen. Topology, 30 (2012), 1–8.
O. De la Cruz, E. J. Hall, P. Howard, K. Keremedis and J. E. Rubin, Unions and the axiom of choice, Math. Log. Quart., 54 (2008), 652–665.
M. Di Nasso and E. Tachtsis, Idempotent ultrafilters without Zorn’s lemma, Proc. Amer. Math. Soc., 146 (2018), 397–411.
D. Fernández-Bretón, Hindman’s theorem in the hierarchy of choice principles, arXiv:2203.06156 (2021).
N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combin. Theory Ser. A, 17 (1974), 1–11.
P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Mathematical Surveys and Monographs Vol. 59, American Mathematical Society (Providence, RI, 1998).
P. Howard and E. Tachtsis, On metrizability and compactness of certain products without the Axiom of Choice, Topology Appl., 290 (2021), 107591.
T. J. Jech, The Axiom of Choice, Studies in Logic and the Foundations of Mathematics Vol. 75, North-Holland Publishing Co. (Amsterdam, 1973).
T. Jech, Set Theory, The Third Millennium Edition, Revised and Expanded, Springer Monographs in Mathematics, Springer-Verlag (Berlin, 2003).
T. Jech and A. Sochor, Applications of the θ-model, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 14 (1966), 351–355.
K. Keremedis and E. Tachtsis, Cellularity of infinite Hausdorff spaces in ZF, Topology Appl., 274 (2020), 107104.
K. Keremedis and E. Wajch, Denumerable cellular families in ZF, Port. Math., 78 (2021), 281–321.
A. Lévy, The independence of various definitions of finiteness, Fund. Math., 46 (1958), 1–13.
D. Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, J. Symb. Logic, 37 (1972), 721–743.
E. Tachtsis, Variants of the principle of dependent choices and a topological proposition, Questions Answers Gen. Topology, 32 (2014), 53–62.
E. Tachtsis, On Ramsey’s theorem and the existence of infinite chains or infinite antichains in infinite posets, J. Symb. Logic, 81 (2016), 384-394.
E. Tachtsis, On the set-theoretic strength of Ellis’ theorem and the existence of free idempotent ultrafilters on ω, J. Symb. Logic, 83 (2018), 551–571.
E. Tachtsis, Łoś’s theorem and the axiom of choice, Math. Log. Quart., 65 (2019), 280–292.
J. Truss, Classes of Dedekind finite cardinals, Fund. Math., 84 (1974), 187–208.
Acknowledgement
We are most grateful to the anonymous referee for careful reading and valuable comments and suggestions which helped us improve the quality and the exposition of the paper. We are especially thankful to the referee for providing us with fruitful information on (the original version of) Hindman’s theorem which resulted in an enhancement of the first paragraph of Section 1 and an enrichment of the bibliography, and for suggesting us a considerable simplification of the original proof of the third claim of Theorem 8.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tachtsis, E. Hindman’s theorem and choice. Acta Math. Hungar. 168, 402–424 (2022). https://doi.org/10.1007/s10474-022-01288-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-022-01288-1
Key words and phrases
- axiom of choice
- weak axioms of choice
- Hindman’s theorem
- Ramsey’s theorem
- chain/anti-chain principle
- permutation models for \(\mathbf{ZFA} + \neg\mathbf{AC}\)
- Pincus’ transfer theorem