In article <firstname.lastname@example.org>, email@example.com wrote:
> On Tuesday, 4 March 2014 21:47:52 UTC+1, Virgil wrote: > > In article <firstname.lastname@example.org>, > > > > email@example.com wrote: > > > > > > > > > But he claimes that the Peano-axioms supply the natural numbers of formal > > > mathematics. So the natural numbers of formal mathematics are: > > > 1, -10, 100, -1000, ... > > > > Elsewhere, the successor operation is denoted by adding one. > > Could you please quote the axiom?
Did I say that it was done in an axiom? But it is quite standard in interpreting the Peano axioms. http://mathworld.wolfram.com/PeanosAxioms.html Peano's Axioms 1. Zero is a (natural) number. 2. If n is a (natural) number, the successor of n is a (natural) number. 3. zero is not the successor of a (natural) number. 4. Two (natural) numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set of (natural) numbers contains zero and also the successor of every (natural) number in it, then every number is in it.
Note that successorship is nowhere defined to be adding one within the axioms themselves. It is a definition made separately.