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Set Theory and its PhilosophyA Critical Introduction$
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Michael Potter

Print publication date: 2004

Print ISBN-13: 9780199269730

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199269730.001.0001

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(p.175) Chapter 11 Ordinals
Set Theory and its Philosophy

Michael Potter (Contributor Webpage)

Oxford University Press

The simple and general principles of induction are powerful tools for proving things about the natural numbers. This chapter investigates ways in which they can be generalized to apply to a very much wider class of ordered sets than the subsets of ω. The basis of this study is the observation that a version of induction can be applied to any ordered set with a property called well-ordering. The strategy is to apply much the same techniques to the study of isomorphism between well-ordered sets used in Chapter 9 to investigate equinumerosity between sets. Just as that work led to an arithmetic of cardinals, what this chapter does is lead to an arithmetic of ordinals.

Keywords:   ordinals, well-ordering, cardinality, transfinite induction, recursion

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