Below is the sketch of the proof I've tried,

Lemma 1. For a transitive set t, .

Proof.

The last step is followed from for a transitive set t.

1. Show that the minimal induction set (a.k.a "a set of natural numbers") is transitive using the above lemma.

For a set theoretic construction of natural numbers, the followings hold (since it is transitive)

.

2. Using a recursion theorem, show that for each , there exists a unique function

(guarantees that it is a function from a set of natural numbers to a set of natural numbers) such that

.

Now, combine 1 & 2 show that if each of m and n is a counting number, then m+n is a counting number.