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Topic: Infinity: The Story So Far
Replies: 16   Last Post: Mar 7, 2014 8:31 AM

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Virgil

Posts: 8,833
Registered: 1/6/11
Re: Infinity: The Story So Far
Posted: Mar 6, 2014 8:07 PM
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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 <4cfd9bb3-5708-4ed7-9285-e2735a4fcd85@googlegroups.com>,
> >>> mueckenh@rz.fh-augsburg.de 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, ...

> >>>
> >>> 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 ur-element 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
--





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