Re: This Month's Thought on Fermat's Last Theorem: 1

From: Randy Poe (poespam-trap_at_yahoo.com)
Date: 10/04/04 

Date: 4 Oct 2004 07:42:01 -0700

Keckman <keckman@welho.com> wrote in message news:<opsfbwg5e03uk9lu@cs81133.pp.htv.fi>...
> Take any 6 natural numbers you want and put them to set.
> What must be the biggest natural number in that set?

There is no limit on what the biggest natural number in
that set must be, since you let me pick any 6.

If you choose {n1, n2, n3, n4, n5, n6}, I can always
find a different set {2*n1, 2*n2, 2*n3, 2*n4, 2*n4,
2*n5, 2*n6} which has a larger maximal element.

> Naturals are not like qeue that's items goes smaller and smaller. "They
> all takes us much place".
>
> n+1 - n = 1
>
> If that's queue's length is supposed to be oo then there is item oo.

OK, I sort of see what you're trying to say. The problem
is that there is a language barrier both with your English
(which I admire you for writing in) and your mathematics
(which is frustrating both of us). It's hard to tell
sometimes whether it is the language or the mathematics
you are confusing.

Anyway, this statement:
> If that's queue's length is supposed to be oo then there is item oo.

is simply wrong. You keep saying it, but it's not true.
That's what you're trying to PROVE, and you "prove" it
by saying it over and over.

You reason thus:

If I have a list {1,...100} of length 100, the maximal
element is 100. I've got 100 different natural numbers,
the larges must be at least 100.

If I have a list {1,..., n} of length n, the maximal element
is n. If I have n different natural numbers, the largest
must be at least n.

If I have a list {1,... } of "length oo", the maximal
element is oo. If I have oo different natural numbers,
the largest must be at least oo.

-------------------------------------

That third does not follow "by induction" from the first
two. The problem is that the list of all natural
number, {1, ...} doesn't have a maximal element. It
is true that if there was one, M with the property
that M was greater than or equal to any natural
number, then M would have to be infinite.

But there isn't such an M in the natural numbers.

There is no requirement that all sets have to have
a largest element. That's what's throwing you. You
think "N is a set, it has a largest element, that
element must be oo, so oo is in N." The second phrase
"it has a largest element" is just not true. No such
axiom.

                 - Randy