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THE RELATION BETWEEN TWO DIMINISHED CHOICE PRINCIPLES

Part of: Set theory

Published online by Cambridge University Press:  15 February 2021

SALOME SCHUMACHER
SALOME SCHUMACHER*
Affiliation:
DEPARTMENT OF MATHEMATICS, ETH ZÜRICH RÄMISTRASSE, 101, 8092 ZÜRICH, SWITZERLANDE-mail: [email protected]
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Abstract

For every $n\in \omega \setminus \{0,1\}$ we introduce the following weak choice principle:

$\operatorname {nC}_{<\aleph _0}^-:$ For every infinite family$\mathcal {F}$ of finite sets of size at least n there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$ with a selection function$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$ such that$f(F)\in [F]^n$ for all$F\in \mathcal {G}$ .

Moreover, we consider the following choice principle:

$\operatorname {KWF}^-:$ For every infinite family$\mathcal {F}$ of finite sets of size at least$2$ there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$ with a Kinna–Wagner selection function. That is, there is a function$g\colon \mathcal {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$ with$\emptyset \not =f(F)\subsetneq F$ for every$F\in \mathcal {G}$ .

We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace $\mathcal {F}$ by a linearly ordered or a well-ordered family.

Keywords

weak forms of the axiom of choiceconsistency resultsFraenkel–Mostowski permutation models of ZFA+ ¬ACPincus’ transfer theoremspartial choice for infinite families of n-element setspartial choice for infinite families of finite sets.

MSC classification

Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E35: Consistency and independence results
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 415 - 432
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Copyright
© The Association for Symbolic Logic 2021

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References

REFERENCES

De La Cruz, O. and Di Prisco, C. A., Weak forms of the axiom of choice and partitions of infinite sets, Set Theory (Di Prisco, C. A., Larson, J. A., Bagaria, J., and Mathias, A. R. D., editors), Springer, Dordrecht, 1998, pp. 47–70.CrossRefGoogle Scholar
De La Cruz, O. and Di Prisco, C. A., Weak choice principles. Proceedings of the American Mathematical Society, vol. 126 (1998), no 3, pp. 867–876.CrossRefGoogle Scholar
Halbeisen, L., Combinatorial Set Theory—With a Gentle Introduction to Forcing, second ed., Springer Monographs in Mathematics, Springer International Publishing, London, 2017.Google Scholar
Halbeisen, L., Plati, R., and Schumacher, S., A new weak choice principle, preprint, 2021, arXiv:2101.07840.Google Scholar
Halbeisen, L. and Schumacher, S., Some implications of Ramsey choice for n-element sets, preprint, 2021, arXiv:2101.06924.Google Scholar
Halbeisen, L. and Tachtsis, E., On Ramsey choice and partial choice for infinite families of n-element sets. Archive for Mathematical Logic, vol. 59 (2020), pp. 583–606.CrossRefGoogle Scholar
Howard, P., Rubin, A. L., and Rubin, J. E., Kinna–Wagner selection principles, axiom of choice and multiple choice. Monatsheft für Mathematik, vol. 123 (1997), pp. 309–319.CrossRefGoogle Scholar
Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Keremedis, K. and Wajch, E., Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in ZF, preprint, 2020, arXiv:2001.00619.Google Scholar
Lévy, A., Axioms of multiple choice. Fundamenta Mathematicae, vol. 50 (1962), no. 5, pp. 475–485.CrossRefGoogle Scholar
Montenegro, C. H., Weak versions of the axiom of choice for families of finite sets, Models, Algebras, and Proofs (Caicedo, X. and Montenegro, C., editors), Lecture Notes in Pure and Applied Mathematics, vol. 203, Dekker, New York, 1999, pp. 57–60.Google Scholar
Pincus, D., Zermelo–Fraenkel consistency results by Fraenkel–Mostowski methods, this Journal, vol. 37 (1972), pp. 721–743.Google Scholar