On Friday, 28 February 2014 06:17:30 UTC+2, Dan Christensen wrote:
> So, this is another axiom?
Of course yes. It's by construction of the natural numbers.
> How about the associativity of + and *? Two more axioms? How about their commutativity? Yet two more axioms? How about their cancelability, etc., etc.....
Those properties are proved after the construction of natural numbers.
Peano's rot does not even construct the natural numbers. It assumes they exist with ALL their properties. The successor function is an assumption of the property that two successive natural numbers have, that is, a unit difference.
Do you honestly think that Peano's "axioms" are anything more than a juvenile attempt on his part? :-)
> I think you will need some equivalent of the Peano's 5th and most important axiom: the principle of induction. You really can't do number theory without it.
You don't Peano's rot. It's part of the natural numbers. :-) For any natural number n, there is always another larger natural number n+1. It's as old as Euclid. Peano did not exist then. :-)