Re: Review of Mueckenheims book.

On Mar 9, 2:38 am, G. Frege <nomail@invalid> wrote:
On 8 Mar 2007 18:23:46 -0800, "MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:
Let's assume that IN only contains finite many elements. Our construction
of IN ensures that all this numbers are finite. Hence there is a biggest
element in IN, say m, and this element is finite. But then m+1 certainly is
also a finite natural number, hence must be in IN, and is bigger than m.
Contradiction!

That's the same contradiction the cranks already refuse to see. What
you say is correct, of course, but we see that your strategy just
comes back around to where we already were with the crank.

Well, you may be right, MoeBlee. I recognized this myself already, hence I
devised a simpler argument for this last step of the argument:

IN (by our "construction") contains ALL and ONLY finite natural
numbers. Hence clearly 0 (or 1 of you like) is in IN. And with
any n in IN certainly n+1 is also in IN (since if n is finite
certainly n+1 is also finite). From this it is clear that IN
does not have an upper bound. Hence IN is an infinite set of
finite natural numbers (since it is growing beyond all bounds
--- which it could not do if it were finite).

Hi, G. Frege.

I haven't read all of the posts you and your interlocutors have made
since I posted my reply to you, but from what you wrote above, I have
to say that again your strategy leads back to a postion we've been in
hundreds and hundreds of times. Unbounded implies infinite. We've been
over and over that with the crank alreday. He doesn't get it.

Of course, this is a "standard consideration", the only "new" aspect (I
think) is the "construction" (definition) of IN mentioned above (starting
with a set IN* which may contain infinite natural numbers, if such numbers
exist). By this construction all infinite numbers have been "tossed out",
and IN only contains finite numbers as elements --- but, as our simple
consideration shows, there must be infinitely many numbers in IN.

This _proves_ that (for example) Mückenheim's claim

"Unless there is an infinite number the number of [natural] numbers
[...] cannot be infinite." (W. Mückenheim)

is simply wrong.

Alas, I'm afraid I don't see how your strategy can help. If your
strategy leads back again to 'unbounded implies infinite', then we're
just right back to where we always are were with the crank.

And remember I mentioned that your strategy depends on getting the
crank to agree that an infinite set less a finite set leaves an
infinite set. It occurred to me that that provokes just another ones
of the crank's variations: The crank thinks that there is such a
cardinality as aleph_0 - 1, which is, for the crank...finite.

We're just continually taken back to the same set of crank variations
on a theme: Unbounded does not imply infinite; aleph_0 - 1 is a
cardinality and it's finite; there is an infinite "process" (Zeno
machine, whatever) that has a last step, etc.

MoeBlee

.

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