Prove: If each of m and n is a counting number, then m+n is a counting number.
A counting number is defined as a number in the minimal induction set.
The minimal induction set is the set of all counting numbers and is given by C.
Lemma 1. For a transitive set t, .
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.