Re: Ultimate debunking of Cantor's Theory

In article <1184550638.931818.34560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Calvin <crice5@xxxxxxxxxxxxxx> wrote:

After finally accepting everything else, I was still
having trouble with the notion that one couldn't use
'not having a largest element' to prove that a set is
infinite, if it is a subset of the naturals, integers,
rationals, or reals, all of which are one-dimensional
number systems.

IF one has a non-empty ordered set without a largest element, one can
easily prove that it satisfies the Dedekind definition of being
infinite, which is that it allows an injection to a proper subset.
This can always be done by allowing a mapping of the members greater
than or equal to some fixed member each to a member strictly larger than
itself, and the identity map on all smaller members.

If one definies finiteness as being bijectable with a natural (or a set
with only a natural number of members) the same construction proves it
not finite.
.

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