Re: Ultimate debunking of Cantor's Theory

Calvin wrote:

One thing that
still bothers me is the claim that showing that a
set does not have a largest element is not a proof of
infinite size.

I understand that plenty of sets without largest
elements are bounded above, but in every case I can
think of (such as the positive rationals strictly
less than 1), the set is infinite. Are there known
counterexamples within the realm of the naturals,
the integers, the rationals, and the reals?

In the case of complex numbers, one can easily
describe finite sets of distinct elements, all having
the same absolute value, and the same would be true
in all such sets of dimension greater than 1.
But for one-dimensional sets, I can't grasp it.


Any *nonempty* *totally* ordered set that does not have a largest element must be infinite. The empty set has no largest element. A partially ordered set with only two elements which are incomparable with each other has no largest element. A nonempty partially ordered set with no *maximal* element must be infinite.

I don't remember where this came up. Can you quote the context?

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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