Definition
52
[1Y1] The axiom of the power set says that for every set \(A\), there is a set \(\mathcal{P}(A)\) whose elements are all and only subsets of \(A\). A shortened definition formula is
\[ {\mathcal P}(A){\stackrel{.}{=}}\bigl\{ B:\ Bβ A\bigr\} \quad . \]
\(\mathcal{P}(A)\) is also called set of parts.
In the formal language of the Zermelo-Fraenkel axioms, the axiom is written:
\[ β A, β\; Z, β y , y β Z \iff (β z, z β y \implies z β A)\quad ; \]
this formula implies that the power set \(Z\) is unique, therefore we can denote it with the symbol \( {\mathcal{P}(A)}\) without fear of misunderstandings.
Note that
\[ (β z, z β y \implies z β A) \]
can be shortened with \(yβ A\) and therefore the axiom can be written as
\[ β A, β\; Z, β y , y β Z \iff ( y β A)\quad ; \]
then using the extensionality, we obtain that
\[ Z=\{ y: ( y β A)\} \quad . \]