1. The definition of natural numbers starts with defining 0 as a number and the sucessor of a number is a number and then uses addition to define n+1 from n.

2.

3. numbers need not have abstract definition as they come intuitively to us as a sentient species.

even if we did not have a definition for the word three in our head we are able to identify the difference between nothing, a unit, a pair, a triplet, and other groups of objects. this is the concept often used to teach babies numbers using dots next to the arabic numbers that we normally use.

4. Originally Posted by talanum1
The definition of natural numbers starts with defining 0 as a number and the sucessor of a number is a number and then uses addition to define n+1 from n.

Yep, the Peano Axioms basically the Natural Numbers and ordinary counting. So what ?

"God made the integers, all else is the work of man." – Leopold Kronecker

5. Originally Posted by talanum1
The definition of natural numbers starts with defining 0 as a number and the sucessor of a number is a number and then uses addition to define n+1 from n.

http://en.wikipedia.org/wiki/Axiom
Welcome to the wonderful world of axiom's.
You will know when you meet one because;
1: They are irreducible.
2: They violate causality, having no predecessor.
3: Without it(the axiom), nothing get's done.

6. You can use the computer and circular numbers to show unsimplifiable theorems are like prime numbers, simplifiable by other unsimplifiable theorems, to their simplest prime state. There is a sieve to find them, etc.

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