Axiom of the Power Set

One of the Zermelo-Fraenkel axioms which asserts the existence for any set a of the power set x consisting of all the subsets of a. The axiom may be stated symbolically as

  forall x exists y( forall z(z in y=z subset x))

(Enderton 1977). Note that the version given by Itô (1986, p. 147),

  forall x exists y(y in x= forall z in y(z in a)),

is confusing, and possibly incorrect.

See also

Power Set, Zermelo-Fraenkel Axioms

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Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146-148, 1986.

Referenced on Wolfram|Alpha

Axiom of the Power Set

Cite this as:

Weisstein, Eric W. "Axiom of the Power Set." From MathWorld--A Wolfram Web Resource.

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