On Wednesday, 5 March 2014 06:22:26 UTC+2, Dan Christensen wrote: > 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?
That's entirely false as anyone who has read my axioms will know. While you can't even define the most simple properties of natural numbers, my axioms based on the brilliance of Ancient Greeks, construct the rational numbers, define the arithmetic and set the foundations for number theory.
You have NOTHING, absolutely nothing but a joke with the spaghetti axioms. :-)
> > > > 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.
The only way is the way I have shown. Everything else is the work of fools.
> > > > So, good luck trying to derive any number-theoretic results whatsoever from these Gobbledygook Axioms of yours, John Gabriel.
Luck has nothing to do with it, only sound and well-thought logic. :-)