Re: Ultimate debunking of Cantor's Theory

"Stephen J. Herschkorn" <sjhersc...@xxxxxxxxxxxx> wrote:
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?

In the 8th post of this thread, Mr. Webb responded
directly to my 2nd post of the thread, as follows:
---------------------------------------------------------
"Calvin" <cri...@xxxxxxxxxxxxxx> wrote:
Proginoskes <CCHeck...@xxxxxxxxx> wrote:

THE ONLY SETS WHICH EXIST ARE FINITE SETS. THE INFINITE IS
ONLY A PRODUCT OF THE IMAGINATION.

Then what is the largest natural number?


I can't see that restricting set theory to finite sets means
there has to be a largest natural number.

So N isn't a set; big deal, nothing breaks. ZFC doesn't have
the set of all ordinals, nobody seems to care.
-----------------------------------------------------------

Later on I tried to say that, by the definition of the
natural numbers, being the same elements as the
sequence, 1,2,3,... (or 1,1+1,1+1+1,...) it follows
that for every n in N (the set of natural numbers), n+1
is also in N; and N couldn't have a largest element L,
because L+1 would have to be in N.

But there were objections to my definition on N, and
to my not having first proved that N is a set.

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.



.

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