Re: Cantor and the binary tree

On 3 Jul 2005 02:39:18 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:

>
>
>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.

Not in ZFC it isn't. That set is not well-founded, and only
well-founded sets exist in ZFC.

>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.

Non sequitor.

Martin

.

Relevant Pages

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