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:

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

| | | | | created: 2014-06-24 21:29:59 | modified: 2018-03-23 00:13:30 | by: *bookofproofs* | references: [656], [983]

## 1.**Corollary**: Justification of Power Set

## 2.**Definition**: Singleton

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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