Re: infinity
- From: Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx>
- Date: Thu, 01 Sep 2005 23:04:54 -0600
In article <1125629551.621618.60990@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"William Hughes" <wpihughes@xxxxxxxxxxx> wrote:
> Tony Orlow (aeo6) wrote:
> > The only contradiction arises from your obsession with a last
> > element, and conflation of it with finiteness for a set. I do not
> > accept that a last element necessarily indicates a finite set,
There are ordered sets with largest elements that are infinite, such as
the closed unit interval, but there are no non-empty ordered sets
without a largest element, at least according to the Cantor definition
of finiteness.
I challenge TO to produce either a set which does not allow any
injections into proper subsets that is infinite by TO's definition or to
produce any set which does allow an injection into a proper subset that
is finite by TO's definition.
Absent TO's successful production of at least one of these two types of
examples, we are free to impose the Cantor criterion for
finiteness/infiniteness of sets on all his postings.
The Cantor criteria are:
A set is finite if and only if there do not exist any injections from
that set to any proper subset;
a set is infinite if and only if there exists at least one injection
from that set to some proper subset.
Note that according to these criteria, a non-empty ordered set without a
maximum (or minimum) member, such as the set of (finite) naturals, is
necessarily infinite.
> > therefore I see no contradiction between the set of finite naturals
> > being finite and not having a last element.
Then TO must have in mind some some definition of finiteness versus
infiniteness of sets incompatible with Cantor's, and should immediately
provide that definition to us so that we can understand what he is
talking about.
If TO cannot provide definitions adequate to support his contentions,
then he must admit that his contentions are unsupportable.
>
> As stated above I realize you believe that there are only a finite
> number of finite integers, and there is no largest finite integer.
> You avoid an explicit contradiction only by refusing to define what
> you mean by infinite. When I said that "TO appears bothered by this
> contradiction" I was referring to your statement "There is no
> well-defined size of this set [the finite integers] despite the fact
> that it must be finite, logically."
>
> -William Hughes
.
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