Re: infinity

In article <MPG.1d879dcdc2d0c1ec98a1e3@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> William Hughes said:
> >
> > Tony Orlow (aeo6) wrote:
> > > 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.
> >
> > 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 refering to your statment "There is no
> > well-defined size of this set [the finite integers] despite the
> > fact that it must be finite, logically."
> >
> > -William Hughes
> >
> >
> Why should a poorly defined set size necessarily be infinite?


If the set is well enough defined that one can show that there is no
fninite upper bound on the size of its members, it is well enough
defined to be proved infinite ( in the sense of having no finite upper
bound).

> What is the contradiction between saying the size is not well
> defined, although it is known to be finite? The number of printed
> words on Earth is also known to be finite, though not a well defined
> number, and without any upper bound.

A rough estimate of the number of atoms making up Earth can be made,
and that number can easily be seen to be an upper bound to the number
of printed words on earth. So "the number of printed words on Earth"
HAS an upper bound!
>
> When I say a number is infinite, one definition might be to say that
> counting to it, using a constant finite unit of time per iteration,
> would take forever.

Linking finiteness/infiniteness to clocks is unsatisfacttory, since
standard set theories are time-independent.

> I am not sure how to defined it to your satisfaction,

Cantor did it nicely to the satisfaction of just about everybody
except TO.

TO will have to come up with something better that Cantor's definition
before anyone will be willing to give up Cantor's definition.


> but I think we all know what we are talking about.

Those who use Cantor's definitions know precisely what they are talking
about. Those who only have only TO's non-definitions have no idea what
anyone is talking about.

> A finite number is one we could count to,

There are lots of finite numbers that we cannot count up to, -1 for
example.

There are even a lot of finite natural numbers that we cannot count to,
as it would take longer than the universe is supposed to have been in
existence to count to them. It was TO's idea t bring time into the
argument.


> and an infinite number is
> greater than any finite number.

How much greater? Circularity!

> I think we agree that if x and y are
> finite, then x+y, x*y, x^y are all finite. So, I am not sure what
> part of my arguments you are missing.

It is the parts of TO's arguments that TO is missing that cause the
problem. To begin, a workable definition of fininteness versus
infiniteness for sets. Cantor has one, TO does not.
.

Relevant Pages

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