
Re: Infinity: The Story So Far
Posted:
Mar 7, 2014 8:28 AM


Virgil wrote: > 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
Does anything exclude S(x) from meaning "x multiplied by 10"? Perhaps there is a problem equating the informal "multiplied by" with the formal *.
 Madam Life's a piece in bloom, Death goes dogging everywhere: She's the tenant of the room, He's the ruffian on the stair.

