World Library  
Flag as Inappropriate
Email this Article

Axiom of dependent choice

Article Id: WHEBN0000441950
Reproduction Date:

Title: Axiom of dependent choice  
Author: World Heritage Encyclopedia
Language: English
Subject: Axiom of choice, Axiom of regularity, List of set theory topics, Set theory, Axiom of countable choice
Collection: Axiom of Choice
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Axiom of dependent choice

In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop most of real analysis.

Contents

  • Formal statement 1
  • Use 2
  • Equivalent statements 3
  • Relation with other axioms 4
  • Footnotes 5
  • References 6

Formal statement

The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (xn) in X such that xnRxn+1 for each n in N. (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way.

If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DCR.

Use

DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step.

Equivalent statements

DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is also equivalent[1] to the Baire category theorem for complete metric spaces.

Relation with other axioms

Unlike full AC, DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.

The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.

Footnotes

  1. ^ Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933--934.

References

  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
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.