On Wednesday, February 26, 2014 4:16:30 PM UTC-5, muec...@rz.fh-augsburg.de wrote:
> > The chracteristic property of the natural numbers is their constant distance of 1 from their next neigbours.
There is nothing about "distance" between numbers in the axioms. The only relation given between numbers is that of succession.
> That feature distinguishes them from all other Peano-sequences. > > > As I said, such structures is embedded in EVERY infinite set. So, you are on fairly safe ground to assume the existence of one such structure at the beginning of a proof. > > > > > But not as the defnition of the natural numbers. The natural numbers can only be defined by the natural process of counting or of adding units, i.e., by the foundation of mathematics.
In a cognitive or historical sense, perhaps. But not in formal mathematics.