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.

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- Feb 22nd 2009, 03:03 PMnoles2188Counting numbers proof
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. - Feb 24th 2009, 03:16 AMaliceinwonderland
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. - Feb 24th 2009, 05:16 AMkalagota
you may also check this:

see this page.. in particular, look on the second post (post by jhevon) and click the hyperlink there...