Re: infinity

William Hughes said:
>
> stephen@xxxxxxxxxx wrote:
> > William Hughes <wpihughes@xxxxxxxxxxx> wrote:
> > > Tony Orlow (aeo6) wrote:
> > >> No, it's really not. This problem is couched as an infinity problem. The
> > >> infinite set of natural numbers requires infinite values. Cantorian thought
> > >> purports to talk about infinity, but then limits itself to finite numbers so as
> > >> to avoid the topic. I said IF you limit yourself to finite numbers, THEN you
> > >> could have an empty vase at noon, although this answer still makes no sense
> > >> given the constantly increasing sum. This is one of the reasons NOT to limit
> > >> the naturals to finite values. There is no well-defined size of this set,
> > >> despite the fact that it must be finite, logically.
> >
> > > I assumed, wrongly, that you accepted the existence of the
> > > finite integers. Your contention that "it [the size of this
> > > set] must be finite, logically", is one of your strangest and
> > > silliest. Why can't there be an infinite set of finite things?
> > > Does the fact that we have an infinite number of ping pong
> > > balls mean some of them must be of infinite size?. Yes, assuming
> > > that there are a finite number of finite integers leads to a
> > > contradiction, as there are clearly an infinite number of them.
> >
> > Tony refuses to precisely define what he means by 'infinite'
> > or 'finite'. Apparently the set of finite integers is finite,
> > or perhaps it is undefined. I think Tony's math allows a set
> > to be neither finite or infinite.
> >
> > Clearly the number of finite integers cannot be a finite
> > integer. Let F be the number of finite integers.
> > Tony agrees that if F is a finite integer, then F+1 is
> > a finite integer. That means that the set {1, 2, 3 ..... F, F+1}
> > contains F+1 finite integers, which contradicts the claim
> > that there were F finite integers.
> >
>
> This uses the fact that a finite set must have a largest element.
> TO (at least implicitely) does not accept this. According
> to TO
>
> -the set of finite integers contains a finite
> number of elements
>
> -there is no largest finite integer
>
> TO appears bothered by this contradiction, his conclusion is that
> the set of finite integers doesn't exist!
>
> > Perhaps in Tonymatics a set can still be finite even
> > if the number of the elements in the set is not finite.
> >
>
> Consistency is not TO's strong suit.
>
> -William Hughes
>
>
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, therefore I see no contradiction between
the set of finite naturals being finite and not having a last element.
--
Smiles,

Tony
.

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