On Thursday, February 27, 2014 6:19:47 PM UTC-5, John Gabriel wrote: > On Friday, 28 February 2014 01:11:55 UTC+2, Dan Christensen wrote: > > > > > Yeah, they pretty much do. > > > > As I explained, you can't use an object in its own definition. The Peano "axioms" assume the prior existence of natural numbers. >
As I explained, in Peano's axioms, we have the essential properties of the natural numbers from which it seems all of their other known properties can be derived (Godel aside). In formal mathematics, these properties can serve as a definition of the natural numbers, especially when you consider that such structures can be found in every infinite set.
You are, of course, free to visualize the natural numbers in any way you find useful, John.
> > > > If your "axioms" aren't somehow equivalent to Peano, they won't go very far. > > > > If my axioms were anything similar to Peano's rot, I would be very concerned!
As I asked WM, can you prove that n=/=n+1 in your system?