The axioms we have introduced so far do not ensure the existence of a power set for a set $X$, containing all the subsets of $X$ as its elements. For instance, the axiom of separation ensures the existence of any subset of $X$ separately, and we could use the axiom of pairing to create a set containing any two of such subsets as elements, but it is not possible to combine all subsets of a given set at once. For this reason, we need another axiom, the axiom of power set.

Axiom: Axiom of Power Set

For each set \(X\) there exists a set containing all subsets of $X$, formally:

$$\forall X~\exists~Y~\forall z~(z\in Y\Rightarrow z\subseteq X).$$


Corollaries: 1

  1. Definition: Singleton

Axioms: 1
Definitions: 2 3
Proofs: 4

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition