On Saturday, March 1, 2014 5:43:02 PM UTC-5, Dan Christensen wrote: > On Saturday, March 1, 2014 2:16:32 PM UTC-5, John Gabriel wrote: > > > On Saturday, 1 March 2014 20:47:36 UTC+2, Dan Christensen wrote: > > > > > > > On Saturday, March 1, 2014 1:31:57 PM UTC-5, John Gabriel wrote: > > > > > > > > > > > > > If you are claiming that number theory can be derived from your "axioms" alone (apart from those of logic and set theory), then it does indeed need a proof. > > > > > > > > > > > > I am not only claiming, but I have shown that number theory can be derived using Euclid's (modified) Elements - call them my axioms if you want. > > > > > > > You have been unable to derive even the most elementary result from number theory. What kind of "axioms" are they? > > > > > > > Axioms don't need proofs. In fact, they usually can't be proved. > > > > > > > The statement, for all natural numbers n, n=/=n+1, is not an axiom (in Peano or in your own system). It is a theorem, the proof of which seems to completely elude you and your system of "axioms." I pick it because it is perhaps the simplest inductive proof in number theory. I have the proof as a worked example in the tutorial than comes with my proof software (see Example 13). >
Apologies. I haven't looked at it in a while, but Example 13 does NOT start from the bare-bones Peano axioms, but rather an enhanced list of fundamental properties of the natural numbers including a definition of addition. Still, it should get you started on a proof using the "bare-bones" axioms.