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)
“The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood.... For the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid.”
—Edgar Allan Poe (1809–1845)
“We should seek by all means in our power to avoid war, by analysing possible causes, by trying to remove them, by discussion in a spirit of collaboration and good will. I cannot believe that such a programme would be rejected by the people of this country, even if it does mean the establishment of personal contact with the dictators.”
—Neville Chamberlain (1869–1940)
“In public buildings set aside for the care and maintenance of the goods of the middle ages, a staff of civil service art attendants praise all the dead, irrelevant scribblings and scrawlings that, at best, have only historical interest for idiots and layabouts.”
—George Grosz (1893–1959)