Re: Cantor and the binary tree

In article <1119734066.741914.71070@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > WMs argument is that any discretely ordered set which contains its
> > bounds (have both a minimum and maximum value within the set) is finite,
> > which is quite true, but that does not apply to discretely ordered sets
> > which do not contain their bounds.
>
> My proof has nothing to do with bounds. The only assertion is that
> every element of the set is an even number which should be satisfied by
> the set of all even numbers.

No you require that each of the sets be finite as well, since they are
all required to be well ordered sets with maximal members. But the union
of all such sets does not have a maximal member.
> >
> > Does WM wish to claim that the set of all even naturals contains a
> > maximum member, a natural greater than any other natural in that set?
>
> Of course it does not. Therefore the cardinal number cannot be larger
> than any member of the set.

WRONG! It means that the cardinality of the set cannot be an even
natural number, nor an odd one either, for that matter.


And the only non-natural cardinalities are the infinite ones.
.

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