Given
- The definition of subset;
- The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$
we know what a subset is and what a power set contains.
In a simple case where a set $A$ is supposed to exist, with $A=\{a, b, c\}$, we know what is and what is not a subset of $A$:
$\{a\}, \{b\}, \{c\}, \{a, b\}, \{a,c\}, \emptyset$ and $A$ are subsets of $A$ and anything different is not.
$\wp(A)=\big\{\{a\}, \{b\}, \{c\}, \{a, b\}, \{b,c\}, \{a,c\}, \{a, b, c\}, \emptyset\big\}$.
However the mere definition of something (and consequently it's recognition as such) does not guarantee its existence. $\emptyset$ and $A$ seem like the only subsets whose existence is immediate.
In other words, I know what a power set contains, but how do I know that the things it contains exist in the first place?
Because such a well-defined and existent set such as $\wp(A)$ should not contain nonexistent elements, to prove the existence of its elements is important. It seems that two alternatives arise:
- Being a member of $\wp(A)$ automatically makes this thing to exist;
or
- There should be an alternative to prove the existence of all subsets of $A$ without the axiom of power set.