Virgil
Posts:
8,833
Registered:
1/6/11


Re: Infinity: The Story So Far
Posted:
Mar 6, 2014 8:07 PM


In article <lf9qbf$so$1@news.albasani.net>, Peter Percival <peterxpercival@hotmail.com> wrote:
> Virgil wrote: > > In article <lf83m8$tqh$1@news.albasani.net>, > > Peter Percival <peterxpercival@hotmail.com> wrote: > > > >> Virgil wrote: > >>> In article <4cfd9bb357084ed79285e2735a4fcd85@googlegroups.com>, > >>> mueckenh@rz.fhaugsburg.de wrote: > >>> > >>>> But he claimes that the Peanoaxioms supply the natural numbers of formal > >>>> mathematics. So the natural numbers of formal mathematics are: > >>>> 1, 10, 100, 1000, ... > >>> > >>> Only in wierd places like WM's wild weird world of WMytheology. > >>> Elsewhere, the successor operation is denoted by adding one. > >> > >> I don't think that is correct. > > > > > > In abstract induction it need not be, but in the standard set of > > naturals, which is what WM is arguing about, it most assuredly is. > > But what are the naturals defined to be? If it's "things satisfying > Peano's axioms" then 1, 10, 100, 1000, ... (with the appropriate S) do > the job.
The natural are defined to be the members of a Peano set, but only when the definitions of addition and multiplication of natural numbers are suitably ( and inductively ) defined, and whether the urelement is interpreted as 0 or 1.
Starting with 0 A x1 in N, 0 S(x1) A x1,x2 in N, S(x1) = S(x2) => x1 = x2 A x1 in N, x1 + 0 = x1 A x1,x2 in N, x1 + S(x2) = S(x1 + x2) A x1 in N, x1 * 0 = 0 A x1,x2 in N, x1 * S(x2) = (x1 *x2) + x1
Starting with 1 A x1 in N, 1 S(x1) A x1,x2 in N, S(x1) = S(x2) => x1 = x2 A x1 in N, x1 + 1 = S(x1) A x1,x2 in N, x1 + S(x2) = x1 + x2 + 1 A x1in N. x1 * 1 = x1 A x1,x2 in N. x1 * S(x2) = (x1 *x2) + x1 

