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.
But "sucessor of" translate into adittion of one and adition already assumes the numbers.

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.
But "sucessor of" translate into adittion of one and adition already assumes the numbers.
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.
Yep, the Peano Axioms basically the Natural Numbers and ordinary counting. So what ?Originally Posted by talanum1
"God made the integers, all else is the work of man." – Leopold Kronecker
http://en.wikipedia.org/wiki/AxiomOriginally Posted by talanum1
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.
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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