Axiom·A04
Power set
The collection of all subsets of a set is itself a set.
In words
For any set A, there is a set P where for every x, x belongs to P exactly when x is a subset of A.
This is a ZFC axiom: it is assumed, not proven. Everything below it in a proof chain ultimately rests here.
Remarks
Guarantees the power set
of D005 exists for every set
. This is the source of the towering hierarchy of ever larger sets and the engine of cardinal comparison: Cantor's theorem shows
is always strictly larger than
.
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