This is the exercise:
(c) Construct a sequence $\{A_n\}_{n \in \Bbb Z_+}$ of infinite sets such that $\forall n \in \Bbb Z_+, A_{n+1} \text{ has cardinality greater than } A_{n}$.
This is the definition of "cardinality greater than":
Let $A.B \neq \emptyset$. We say that $A$ has cardinality greater than $B$ if and only if there exists an injection mapping $B$ into $A$ and there is no injection mapping $A$ into $B$.
My attempt
The natural choice is letting $A_1 = \Bbb Z_+$ and $A_{n+1} = \mathcal{P}(A_n)$, where $\mathcal{P}$ denotes the power-set. Now, my problem is that in order to do so we must define a function $h$, by the Principle of Recursive Definition, such that:
$$ \begin{align} h(1) &= A_1 \\ h(n+1) &= \mathcal{P}(A_n) \end{align}$$
however I am missing the codomain of such function (without which the definition is not well-posed) and I do not know how to construct it from the Principles of Naive Set Theory.
As always, any hint or comment is welcome and let me know if I can explain myself clearer!