In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:
Or, if we've already defined the subset operation:
Or in words:
Given anysetA, there is a set B such that, given any set C, C is a member of Bif and only if, given any set D, ifD is a member of C, then D is a member of A.
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that C is a subset of A. Thus, what the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the subsets of A. We can use the axiom of extensionality to show that this set B is unique. We call the set B the power set of A, and denote it PA. Thus the essence of the axiom is:
Every set has a power set.
The axiom of power set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.
Cantor's diagonal argument shows that the powerset of a set (infinite or not) always has strictly higher cardinality than the set itself (informally the powerset must be 'greater' than the original set).
The powerset of the set of natural numbers for instance can be put in a one-to-one correspondence with the set of real numbers (by identifying an infinite 0-1 sequence with the set of indices where the ones occur).
The powerset of a set S forms an Abelian group when considered with the operation of symmetric difference (with the empty set as its unit and each set being its own inverse) and a commutative semigroup when considered with the operation of intersection.
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