Re: Cantor and the binary tree



Virgil wrote:

> > Of course every set of naturals is bounded by a natural. What else
> > should be in there? A television set? We cannot determine the magnitude
> > of it, but we know that it is finite and is maximum of its initial
> > sequence.
>
> "We" don't know any such thing. In fact, the Peano postulates
> specifically forbid the existence os any such thing.

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. This segment is potentially infinite, because there is no
largest n and because it can be mapped on a proper subset.
Nevertheless, it is not actually infinite.
I am sure you would be intelligent enough to understand that, unless
set theory had spoiled the minds of you to a large extent.


> 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.

Regards, WM

.

Relevant Pages

  • Re: Cantor and the binary tree
    ... >>> Of course every set of naturals is bounded by a natural. ... > a pair of different elements n and n' of N, which satisfy the following ... > n belongs to an initial segment S which does not contain n' ... > n' belongs to an initial segment S' which does not contain n ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >>> Of course every set of naturals is bounded by a natural. ... >a pair of different elements n and n' of N, which satisfy the following ... >n belongs to an initial segment S which does not contain n' ... >n' belongs to an initial segment S' which does not contain n ...
    (sci.math)
  • 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 and the binary tree
    ... >>> Every countable set like N is potentially infinite but not actually ... > n belongs to an initial segment S which does not contain n' ... > n' belongs to an initial segment S' which does not contain n ... Not so (unless WM insists that N is a member of itself, ...
    (sci.math)
  • Phool nearly defines multiplication
    ... 2-unit segment, and if we equate the points in each ... one set could have twice as many numbers ... The ODD naturals are the SAME size as ALL the naturals! ... Yet for n in the ODD naturals, 2n is NOT in the Odd naturals! ...
    (sci.logic)
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