Re: infinity

In article <MPG.1d879fe6c1c5af7398a1e5@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > 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


> All whole numbers from 100...000 through 111...111.

Now PROVE that there is no injection from the set of all those whatever
they are to any proper subset!

> > 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.

> Yes, Cantor-infinite, as you say, but not infinite by other thinking.

Cantor-infinite is the only infinite until we have a workable
alternative. So far none of TO's attempts have been workable
> >
> > > > 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.

> The rules of finiteness that you seem to be ignoring are that a+b,
> a*b and a^b are all finite for finite a and b. Do you disagree with
> that statement?

If a and b are any members of any ring, including the ring of integers
or the field of reals, a+b and a*b will be members of that same ring,
but a^b need not be.

Since "standard" rings are of finite elements (their compactifications
are not, in general, rings), the sum or product of two members will be
as finite as their summands or factors.
.

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