# Why is the Axiom of Power Set needed?

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## Main Question or Discussion Point

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

Axiom. Given any set x, there is a set
such that, given any set z, this set z is a member of
if and only if every element 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.

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fresh_42
Mentor
In the Zermelo-Fraenkel axioms of axiomatic set theory we find:

Axiom. Given any set x, there is a set
such that, given any set z, this set z is a member of
if and only if every element 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.
Definitions are only names. They do not have any other function than to support language and abbreviate it.
Axioms are rules. They determine whether a conclusion is allowed or not.

Why does a power set exist at all? Do you know a derivation from the other axioms?

If it was a definition, then it would read: "A set with this and that property is called a power set."
As an Axiom, the existence of such a set is required: "There is a power set."
So the axiom frees you from the need to construct one, which would be rather difficult for let's say $x=\mathbb{R}\,.$

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

Axiom. Given any set x, there is a set
such that, given any set z, this set z is a member of
if and only if every element 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.
Apparently it is used to prove that Cartesian products exist. Look at the proof used here where there is no such axiom.

Math_QED
Homework Helper
2019 Award
You can only define objects that already exist.

As an easy example, we can define $\sqrt{x}$ for $x \geq 0$ as the unique number $y \geq 0$ with $y^2 =x$.

But who says that such a number $y$ exists? And that it is unique? These things have to be verified (and any proof will somehow invoke the LUB property), or the definition wouldn't make sense.

For the same reason, before we define what the term "power set" means, we have to verify if that object exists.

Since we can't deduce the existence of the power set from the other ZFC axioms, it is added as an axiom.

mathwonk