World Library  
Flag as Inappropriate
Email this Article

Successor ordinal

Article Id: WHEBN0000375256
Reproduction Date:

Title: Successor ordinal  
Author: World Heritage Encyclopedia
Language: English
Subject: Ordinal arithmetic, Limit ordinal, Reference desk/Archives/Mathematics/2015 February 20, Transfinite induction, Regular cardinal
Collection: Ordinal Numbers
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal.

Contents

  • Properties 1
  • In Von Neumann's model 2
  • Ordinal addition 3
  • Topology 4
  • See also 5
  • References 6

Properties

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.[1]

In Von Neumann's model

Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor S(α) of an ordinal number α is given by the formula[1]

S(\alpha) = \alpha \cup \{\alpha\}.

Since the ordering on the ordinal numbers α < β if and only if α ∈ β, it is immediate that there is no ordinal number between α and S(α), and it is also clear that α < S(α).

Ordinal addition

The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:

\alpha + 0 = \alpha\!
\alpha + S(\beta) = S(\alpha + \beta)\!

and for a limit ordinal λ

\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)

In particular, S(α) = α + 1. Multiplication and exponentiation are defined similarly.

Topology

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.[2]

See also

References

  1. ^ a b Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46,  .
  2. ^ Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory,  .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.