Abstract
This chapter introduces the basic idea of cardinal numbers, comparability, and operations, and next covers the theory of finite sets and natural numbers, from which the Dedekind–Peano axioms are derived as theorems. Dedekind infinite sets and reflexive cardinals are also defined. It then presents the axiom of!choiceAxiom of Choice and contrasts it with effective!choiceeffective choice, using the notion of effectivenesseffectiveness informally. The rest of the chapter is about countability and uncountability: It focuses on the two specific cardinals ℵ 0 = | N | and \(\boldsymbol{\mathfrak{c}} = \vert \mathbf{R}\vert \), and gives the first proof of \(\aleph _{0} <\boldsymbol{ \mathfrak{c}}\) (uncountability of \(\mathbf{R}\)). In the process, the principles of countable axiom of choice (CAC)countable and axiom of!dependent choice (DC)dependent choice are encountered.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that if \(\mathcal{P}\) is finite, the sets in \(\mathcal{P}\) may be infinite. We should be careful to distinguish between “a partition being finite” and “the sets in the partition being finite.” For example, the partition of the set of integers into even and odd integers is a finite partition consisting of infinite sets, while the partition \(\{\{2n, 2n + 1\}\!\mid n \in \mathbf{Z}\}\) of the integers is an infinite partition consisting of finite sets.
- 2.
Effectiveness is a metamathematical notion, and degrees of effectiveness (which depends on the complexity of the specification or rule) are studied in areas of mathematical logic such as recursion theory.
- 3.
This is the coset decomposition of the additive group \(\mathbf{R}\) modulo the subgroup \(\mathbf{Z}.\)
- 4.
This is the coset decomposition of the additive group \(\mathbf{R}\) modulo the subgroup \(\mathbf{Q}\), and is sometimes called the Vitali partition.
- 5.
AC is needed even if the sets A m are all finite, as illustrated by Russell’s example: Given ℵ 0 pairs of socks and ℵ 0 pairs of boots, how many socks do we have in total, and how many boots? With boots, the answer is ℵ 0, but the socks may form an infinite Dedekind finite set and the answer may be a non-reflexive cardinal.
References
K. Kuratowski. Topology, volume I. Academic Press, 1966.
B. Russell. Introduction to Mathematical Philosophy. Public Domain (originally published by George Allen & Unwin and Macmillan), 2nd edition, 1920.
E. Zermelo. Proof that every set can be well-ordered (1904). In From Frege to Gödel [78], pages 139–141. English translation of “Beweis, daß jede Menge wohlgeordnet werden kann”, Mathematische Annalen, 59(4):514–516, 1904.
E. Zermelo. A new proof of the possibility of a well-ordering (1908). In From Frege to Gödel [78], pages 183–198. English translation of “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Mathematische Annalen, 65(1):107–128, 1908.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dasgupta, A. (2014). Cardinals: Finite, Countable, and Uncountable. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_5
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8854-5_5
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4614-8853-8
Online ISBN: 978-1-4614-8854-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)