- By axiom of infinity, we have a set.
- By axiom of specification, we can create all of its subsets.
- By axiom of pairing and axiom of unions, we can create a set containing all of theses created subsets.
Example: Suppose a set $S$ has $a,b$ and $c$ as its only distinct subsets. (Well it isn’t possible, but for the time’s sake let it be. It is also applicable for correct number of subsets.) Now, by pairing we have $\{ a,b\}$ and $\{ c\}$. By pairing again, we have $\{ \{ a,b\} ,\{ c\} \}$. By union, we have $\{ a,b,c\}$.
We have thus constructed the power set of the given set.
Why then do we need axiom of powers?