Why is the Axiom of Power Set needed?

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

Axiom. Given any set x, there is a set
dcde29bc2e45577cf48fce37eace431df129adf0
such that, given any set z, this set z is a member of
dcde29bc2e45577cf48fce37eace431df129adf0
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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  • #2
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In the Zermelo-Fraenkel axioms of axiomatic set theory we find:

Axiom. Given any set x, there is a set
dcde29bc2e45577cf48fce37eace431df129adf0
such that, given any set z, this set z is a member of
dcde29bc2e45577cf48fce37eace431df129adf0
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}\,.##
 
  • #3
verty
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In the Zermelo-Fraenkel axioms of axiomatic set theory we find:

Axiom. Given any set x, there is a set
dcde29bc2e45577cf48fce37eace431df129adf0
such that, given any set z, this set z is a member of
dcde29bc2e45577cf48fce37eace431df129adf0
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.
 
  • #4
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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.
 
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mathwonk
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i am a novice in this area, but in my opinion an essenctial point here is to note that the word "set" has a special significance in ZF theory. I.e. to avoid problems arising from naive treatment of sets, one has to disallow certain collections of objects that can be easily described, from being called "sets". So even though the description of P(X) seems to be something one could easily imagine, hence it surely exists in the imagination, it still is saying something that this object deserves to be labeled a "set". I.e. those classes of objects that are called sets must be restrictive enough so as not to allow any of the usual ("set of all sets") paradoxes to arise. The idea here seems to be that if one has a class of allowable sets that do not lead to a paradox, then including also all their powers, then still no paradox will arise. Without that, this axiom would not be a good one. I.e. the collection of all sets appears also to exist, at least in language, but it apparently may not be called a "set" in ZF theory. So, building on the previous post, an axiom is added when it not only gives something new and useful, but (hopefully) also does not lead to a contradiction.
 

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