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Number, successor, zero are terms which we presume to know the meaning of.

We then use five propositions:

1. 0 is a number.

2. Every number has a successor.

3. 0 is not the successor of any number.

4. Any proeprty common to zero and its successor and its successors successors is a property of all numbers. (basically any property common to zero, 1 and 2 is a property of all numbers)

5. No two numbers have the same successor.

I started with M + 0 = M

and M + 1 = successor of M.

and for multiplication M * N = M + M ....(n times)