On Wednesday, February 26, 2014 11:58:07 AM UTC-5, John Gabriel wrote:
> > > > > In your write-up, next(a) is meaningless garbage. > > > > > It is a successor function on the set of natural numbers. Got a problem with that? > > > > Yes, because then you already assume the prior existence of natural numbers. > > > Again, unless you want to rule out the existence of infinite sets altogether, the existence of the natural numbers is not problematic. See my previous posting above. > > > This is what you claim: > > > > The Definition of the set of natural numbers is given by nothing more or less than Peano's Axioms. > > > > It is false. The Peano axioms assume the prior existence of natural numbers.
The Peano axioms describe structures that are embedded in EVERY infinite set.
> To with, the next(a) function you refer to in your statements depends not only on the prior existence of natural numbers, but also on the well ordering.
It depends only on the existence of at least one infinite set. Do you claim they don't exist? That is certainly one school of thought, far outside the mainstream though it may be.