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
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 power set

From Wikipedia, the free encyclopedia

The elements of the power set of the set {x, y, z} ordered with respect to inclusion.

In mathematics, the axiom of power set[1] is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set the existence of a set , the power set of , consisting precisely of the subsets of . By the axiom of extensionality, the set is unique.

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

YouTube Encyclopedic

  • 1/5
    Views:
    1 037
    4 036
    2 970
    18 364
    15 548
  • What is the Power Axiom? (Axiom 6 Set Theory)
  • Zermelo Fraenkel Powerset
  • (Axiomatic Set Theory, 5) Power Set Axiom and Axiom Schema of Comprehension
  • 1.11.11 Set Theory Axioms: Video [Optional]
  • Zermelo-Fraenkel Set Theory

Transcription

Formal statement

The subset relation is not a primitive notion in formal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation is defined in terms of set membership, . Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set reads:

where y is the power set of x, z is any element of y, w is any member of z.

In English, this says:

Given any set x, there is a set y such that, given any set z, this set z is a member of y if and only if every element of z is also an element of x.

Consequences

The power set axiom allows a simple definition of the Cartesian product of two sets and :

Notice that

and, for example, considering a model using the Kuratowski ordered pair,

and thus the Cartesian product is a set since

One may define the Cartesian product of any finite collection of sets recursively:

The existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.

Limitations

The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do.[2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set theory could contain sets that are not constructible.

References

  1. ^ "Axiom of power set | set theory | Britannica". www.britannica.com. Retrieved 2023-08-06.
  2. ^ Devlin, Keith (1984). Constructibility. Berlin: Springer-Verlag. pp. 56–57. ISBN 3-540-13258-9. Retrieved 8 January 2023.
  • Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
  • Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This page was last edited on 22 March 2024, at 21:31
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.