Re: Cantor and the binary tree

In article <1120322963.884135.286850@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > >
> > > Every countable set like N is potentially infinite but not actually
> > > infinite
> >
> >
> > Then WM is only potentially numerate but not actually numerate.
>
> Look here: The natural number n e N is nothing else than an
> abbreviation of its initial segment {1,2,3,...,n} c N.
> N consists exclusively of elements n. Similarly N consists exclusively
> of subsets = initial segments (all of which include 1). There is no
> element of N which is not an element of such a subset. And there is not
> a pair of different elements n and n' of N, which satisfy the following
> condition:
> n belongs to an initial segment S which does not contain n'
> and
> n' belongs to an initial segment S' which does not contain n
> in short:
> n e S and n' !e S and n' e S' and n !e S'.
> As this requirement is impossible to satisfy, the segment of n includes
> all elements less than n. This holds for any n e N. Therefore N is a
> segment.

Not so (unless WM insists that N is a member of itself, which situation
is prohibited by most set theories). N not being a member of itself
means N does not have to have any property that its members have.


\
>
> > WM's problem is that a set that is only potential is not a set at all.
> > Sets must be well defined. Whatever it is that WM is describing is not a
> > well defined set.
>
> There are no well defined sets other than finite sets.

Then there is no such thing as a set of natural number or a set of
rational numbers or a set of real numbers, and all of arithmetic must be
thrown out.

WM conflates the theory (pure mathematics) with the practice (science).
>
> Regards, WM
.

Relevant Pages

  • Re: Cantor and the binary tree
    ... > Then WM is only potentially numerate but not actually numerate. ... n belongs to an initial segment S which does not contain n' ... n' belongs to an initial segment S' which does not contain n ... This segment is potentially infinite, ...
    (sci.math)
  • Re: Cantor Confusion
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  • Re: Cantor and the binary tree
    ... >> Of course every set of naturals is bounded by a natural. ... n belongs to an initial segment S which does not contain n' ... n' belongs to an initial segment S' which does not contain n ... This segment is potentially infinite, ...
    (sci.math)
  • Re: Cantor Confusion
    ... N is a linear set (every finite initial segment (= line) includes ... The potentially infinite sequence of natural numers is an initial ... Every natural number belongs to the sequence and to at least one ...
    (sci.math)
  • Re: Cardinality of Set of Computable Numbers?
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