To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Axiom of global choice

From Wikipedia, the free encyclopedia

In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set.

YouTube Encyclopedic

  • 1/3
    Views:
    999
    453
    1 591
  • Axiomatic Theories of Truth (Typed Theories)
  • Values of L-Functions and Modular Forms - Chris Skinner
  • Judaism and Economics- Professor Uzi Segal

Transcription

Statement

The axiom of global choice states that there is a global choice function τ, meaning a function such that for every non-empty set z, τ(z) is an element of z.

The axiom of global choice cannot be stated directly in the language of Zermelo–Fraenkel set theory (ZF) with the axiom of choice (AC), known as ZFC, as the choice function τ is a proper class and in ZFC one cannot quantify over classes. It can be stated by adding a new function symbol τ to the language of ZFC, with the property that τ is a global choice function. This is a conservative extension of ZFC: every provable statement of this extended theory that can be stated in the language of ZFC is already provable in ZFC (Fraenkel, Bar-Hillel & Levy 1973, p.72). Alternatively, Gödel showed that given the axiom of constructibility one can write down an explicit (though somewhat complicated) choice function τ in the language of ZFC, so in some sense the axiom of constructibility implies global choice (in fact, [ZFC proves that] in the language extended by the unary function symbol τ, the axiom of constructibility implies that if τ is said explicitly definable function, then this τ is a global choice function. And then global choice morally holds, with τ as a witness).

In the language of von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory, the axiom of global choice can be stated directly (Fraenkel, Bar-Hillel & Levy 1973, p.133), and is equivalent to various other statements:

In von Neumann–Bernays–Gödel set theory, global choice does not add any consequence about sets (not proper classes) beyond what could have been deduced from the ordinary axiom of choice.

Global choice is a consequence of the axiom of limitation of size.

References

  • Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of set theory, Studies in Logic and the Foundations of Mathematics, vol. 67 (Second revised ed.), Amsterdam-London: North-Holland Publishing Co., ISBN 978-0720422702, MR 0345816
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
  • John L. Kelley; General Topology; ISBN 0-387-90125-6
This page was last edited on 6 March 2024, at 07:25
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.