On Tuesday, March 4, 2014 6:31:59 PM UTC-5, Dan Christensen wrote: > On Tuesday, March 4, 2014 6:05:43 PM UTC-5, John Gabriel wrote: > > > On Tuesday, 4 March 2014 21:04:22 UTC+2, Dan Christensen wrote: > > > > It is an essential property of the natural numbers: there exists at least one element of N. > > > It has nothing to do with natural numbers. > > Well, you go ahead play around with number systems without any actual numbers in them, and see how far you get, John Gabriel. Sounds really boring to me. >
Note, too, that the only constants you give in your "axioms" are a '0' and something called a "unit". Unlike the case with the Peano axioms, nowhere are these constants called numbers. Why is that, John Gabriel?
The only way to create new numbers is applying difference, sum, reciprocal and multiply operators to existing numbers. If you start with no numbers, you cannot generate new ones even if these operators were well-defined -- which, as we have seen here, they are not.
So, good luck trying to derive any number-theoretic results whatsoever from these Gobbledygook Axioms of yours, John Gabriel.