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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: 2019-08-03 19:21:30 | by: bookofproofs | references: [656], [983]

1.Corollary: Justification of Power Set

2.Definition: Singleton

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[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