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].
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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.
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Tachtsis, E. Hindman’s theorem and choice. Acta Math. Hungar. 168, 402–424 (2022). https://doi.org/10.1007/s10474-022-01288-1
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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