## Notre Dame Journal of Formal Logic

### Finiteness Classes and Small Violations of Choice

#### Abstract

We study properties of certain subclasses of the Dedekind finite sets (addressed as finiteness classes) in set theory without the axiom of choice (AC) with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows:

1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of its elements are comparable.

2. The principle “Small Violations of Choice” (SVC)—introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above.

3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF.

4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above.

5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 3 (2016), 375-388.

Dates
Accepted: 7 November 2013
First available in Project Euclid: 26 February 2016

https://projecteuclid.org/euclid.ndjfl/1456499492

Digital Object Identifier
doi:10.1215/00294527-3490101

Mathematical Reviews number (MathSciNet)
MR3521487

Zentralblatt MATH identifier
06621296

Subjects
Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E35: Consistency and independence results

#### Citation

Herrlich, Horst; Howard, Paul; Tachtsis, Eleftherios. Finiteness Classes and Small Violations of Choice. Notre Dame J. Formal Logic 57 (2016), no. 3, 375--388. doi:10.1215/00294527-3490101. https://projecteuclid.org/euclid.ndjfl/1456499492

#### References

• [1] Blass, A., “Injectivity, projectivity, and the axiom of choice,” Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31–59.
• [2] De la Cruz, O., “Finiteness and choice,” Fundamenta Mathematicae, vol. 173 (2002), pp. 57–76.
• [3] Felgner, U., “Choice functions for sets and classes,” pp. 217–55 in Sets and Classes (On the Work by Paul Bernays), edited by G. H. Müller, vol. 84 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1976.
• [4] Felgner, U., and T. J. Jech, “Variants of the axiom of choice in set theory with atoms,” Fundamenta Mathematicae, vol. 79 (1973), pp. 79–85.
• [5] Herrlich, H., “The finite and the infinite,” Applied Categorical Structures, vol. 19 (2011), pp. 455–68.
• [6] Herrlich, H., and E. Tachtsis, “On the number of Russell’s socks or $2+2+2+\cdots=$?,” Commentationes Mathematicae Universitatis Carolinae, vol. 47 (2006), pp. 707–17.
• [7] Howard, P. E., A. L. Rubin, and J. E. Rubin, “Independence results for class forms of the axiom of choice,” Journal of Symbolic Logic, vol. 43 (1978), pp. 673–84.
• [8] Howard, P. E., and J. E. Rubin, Consequences of the Axiom of Choice, vol. 59 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 1998.
• [9] Howard, P. E., and M. F. Yorke, “Definitions of finite,” Fundamenta Mathematicae, vol. 133 (1989), pp. 169–77.
• [10] Jech, T. J., The Axiom of Choice, vol. 75 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973.
• [11] Jech, T. J., Set Theory, 3rd millennium edition, revised and expanded, Springer, Berlin, 2002.
• [12] Kunen, K., Set Theory: An Introduction to Independence Proofs, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1980.
• [13] Lévy, A., “The independence of various definitions of finiteness,” Fundamenta Mathematicae, vol. 46 (1958), pp. 1–13.
• [14] Tarski, A., “Sur les ensembles finis,” Fundamenta Mathematicae, vol. 6 (1924), pp. 45–95.
• [15] Truss, J., “Classes of Dedekind finite cardinals,” Fundamenta Mathematicae, vol. 84 (1974), pp. 187–208.