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## Main Question or Discussion Point

In the Zermelo-Fraenkel axioms of axiomatic set theory we find:

Axiom. Given any set
such that, given any set
if and only if every element of

Why is this needed as an axiom? why isn't it merely a definition? Under what situation would the existence of the power set be in question? seems like it can always be constructed.

Axiom. Given any set

*x*, there is a set*z*, this set*z*is a member of*z*is also an element of*x*.Why is this needed as an axiom? why isn't it merely a definition? Under what situation would the existence of the power set be in question? seems like it can always be constructed.