, 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:
- ∀ A, ∃ B, ∀ C, C ∈ B ↔ (∀ D, D ∈ C → D ∈ A);
or in words:
- Given any set A, there is a set B such that, given any set C, C is a member of B if and only if, given any set D, if D 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
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
We call the set B
the power set
, 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.