Math and science::Analysis::Tao::02. The natual numbers
Tao's 5 axioms for natural numbers1. 0 is a natural number.
The root node.
2. If n is a natural number, then n++ is also a natural number.
The increment operation. The only axiomatic operation.
3. If \( n++ = m++ \) for any natural numbers n and m, then \( n = m \). (Different natural numbers must have different successors).
Prevent loops where there are two nodes that increment to the same node.
4. \( n++ \ne 0 \) for every natural number n. (0 is not the successor of any natural number).
Prevent a circle loop back to 0 (not covered by 3).
5. Let P(n) be any property pertaining to a natural number. Suppose that P(0) is true, and suppose that whenever P(n) is true P(n++) is also true. Then P(n) is true for every natural number n.
There are no other connected components other than the one containing 0. A stange approach to address this. Tao mentions that this is more of an 'axiom schema' in logic parlance.