Assioma dell'insieme potenza

In matematica, l'assioma dell'insieme potenza è uno degli assiomi della teoria degli insiemi di Zermelo-Fraenkel.

Nel linguaggio formale degli assiomi di Zermelo-Fraenkel, l'assioma si scrive:

Oppure a parole:

Dato un generico insieme A, esiste un insieme tale che, dato un generico insieme B, B è un elemento di se e solo se B è un sottoinsieme di A.

Per l'assioma di estensionalità questo insieme è unico. Chiamiamo l'insieme insieme potenza di A. Quindi l'essenza dell'assioma è:

Ad ogni insieme corrisponde un insieme potenza.

L'assioma dell'insieme potenza è generalmente considerato non controverso, e appare in questa forma o in una forma equivalente in quasi tutte le assiomatizzazioni alternative della teoria degli insiemi.

Conseguenze

L'assioma dell'insieme potenza permette la definizione del prodotto cartesiano di due insiemi e :

Il prodotto cartesiano è un insieme dal momento che

Si può definire il prodotto cartesiano di ogni collezione finita di insiemi ricorsivamente:

Collegamenti esterni

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