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

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Posts: 8,833
Registered: 1/6/11
Re: Infinity: The Story So Far
Posted: Mar 5, 2014 3:24 PM
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In article <>, wrote:

> On Tuesday, 4 March 2014 23:01:44 UTC+1, Virgil 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.

> My observation is that this standard interpretation is nothing that follows
> from the axioms.

Then, as usual, WM is being extremely careful not to observe the obvious.

> It follows from the blindness of those who think that these
> axioms define the natural numbers or that axioms are a necessary starting
> point for mathematics.

The axioms certainly charactize the set of natural numbers and any other
set suitable for induction, and, absent induction, WM cannot derive the
standard arithmetic of those natural numbers or any other standard set
of numbers erived from the naturals.

What axioms does WM use to justify inducton if not something so very
like the Peano axioms as to be logically indistinguishable from them?
> >
> > Note that successorship is nowhere defined to be adding one within the
> > axioms themselves. It is a definition made separately.

> Note that these few axioms define the natural numbers:
> 1 in M
> If n in M then n + 1 in M
> The elements of |N are in every such M.

How is this not an easy consequnceof the Peano axioms?
> Of course you must know what + 1 means. But if you don't then you need no
> numbers at all. Similarly it is valid for all axioms: If you can't read or
> think then you need no axioms at all.

And WM apparently needs none in his WMytheology. .

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