Re: infinity


Tony Orlow wrote:
> Daryl McCullough said:
> > Tony Orlow says...
> >
> > >> A sequence of length aleph_0 has a domain equal to the set of all
> > >> ordinals less than aleph_0.
> > >So, you are placing aleph_0 directly after the largest finite?
> >
> > How many times do you have to be told: There *is* no largest
> > finite! aleph_0 is placed after *all* finite ordinals.
> And has no predecessor. Good way to kludge away the largest finite.

You seem to have changed from vehememtly denying the existence
of the largest finite, to vehemently proclaiming the existence.
However, if one takes the position that there is no largest finite,
then there is no need to kludge it away.



> >
> > Here's a way to visualize countable ordinals that might
> > make sense to you.
> >
> > Take the real number line, and label the point 0 with the
> > ordinal 0. Label the point 1/2 with the ordinal 1. Label
> > the point 2/3 with the ordinal 2. In general, label the
> > point n/(n+1) with the ordinal n. Finally, label the
> > point 1 with the ordinal aleph_0.
> The point 1 is not included unless you allow infinite n, right?

Right. And of course aleph_0 is just such an infinite n
(you are keeping in mind the fact that we are talking about
ordinals, not naturals).

> >
> > Note: the finite ordinals are the labels for reals of
> > the form n/(n+1). There is no largest real of that form,
> > but the real 1 is larger than *any* of them.
> When n=oo.

There is no "n=00". We are talking about "finite ordinals"
so we only have finite naturals. Note
TO infinite naturals will not work here as for any
TO natural (finite or infinite) n/(n+1) does not equal 1.

> >
> > >That's what it seems like. Each of those sets you mention
> > >has a largest element 1 less than its size.
> >
> > Right. That pattern holds for every finite set. That's
> > because finite sets of naturals have a largest element.
> > Infinite sets of naturals *don't* have a largest element.
> But then, the largest of that set would be one less than its size, which is
> aleph_0. Does that mean aleph_0 is one greater than the largest finite? If not,
> why not?

You are kind of stuck on the idea that all sets have a largest element.
You take the statement

Infinite sets of naturals *don't* have a largest element

and reply

But then, the largest of that set...

If a set doesn't have a largest element, "the largest of
that set" does not exist. It follows that it is not true
that for every set "the largest of the set is one less than
its size". So there is no reason to assume that one less
than aleph_0 makes any sense. (In fact it does not).

- William Hughes

.

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