Natural numbers defined as cardinals of finite sets?

Question

I am interesting on the formal construction of $\mathbb{N}$. Obviously, I am familiarized with Peano axioms. I am looking for another constructions. For example, using Zermelo-fraenkel axioms, one can construct $\mathbb{N}$ as the intersection of all the inductive subsets of an inductive subset (which exits by virtue of the axiom of infinity). My question is if this could be another way to construct natural numbers.

1. Define equipotence relation over the class of all sets.
2. Define finite sets.
3. The axiom of cardinality says that for each set A there exists a unique cardinal $a$ such that $a$ is equipotent to A.
4. Define $\mathbb{N}$ as the set L of all finite cardinal numbers (note that a cardinal is a set and we know when a set is finite or not).

Is this a correct way to define natural numbers? How can be shown that L is indeed a set?

Thanks in advance.