Axiom of Power Set

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

where P stands for the power set of A, . In English, this says:

Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)

By the axiom of extensionality this set is unique. We call the set the power set of A. Thus, the essence of the axiom is that every set has a power set.

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.

Read more about Axiom Of Power Set:  Consequences

Famous quotes containing the words axiom of, axiom, power and/or set:

    It’s an old axiom of mine: marry your enemies and behead your friends.
    —Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)

    It’s an old axiom of mine: marry your enemies and behead your friends.
    —Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)

    Every diminution of the public burdens arising from taxation gives to individual enterprise increased power and furnishes to all the members of our happy confederacy new motives for patriotic affection and support.
    Andrew Jackson (1767–1845)

    I have resolved on an enterprise that has no precedent and will have no imitator. I want to set before my fellow human beings a man in every way true to nature; and that man will be myself.
    Jean-Jacques Rousseau (1712–1778)