Abstract
This chapter concludes our development of cardinals and ordinals. We introduce the first uncountable ordinal, the alephs and their arithmetic, Hartogs’ construction, Zermelo’s well-ordering theorem, the comparability theorem for cardinals, cofinality and regular, singular, and inaccessible cardinals, and the Continuum Hypothesis.
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.
From the standard axioms for set theory (such as ZFC), assuming they are consistent; this follows from a result of Gödel known as the second incompleteness theorem.
- 2.
It was the first in Hilbert’s celebrated list of problems presented in 1900.
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). Alephs, Cofinality, and the Axiom of Choice. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_10
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8854-5_10
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)