Re: infinity


Tony Orlow (aeo6) wrote:
> William Hughes said:
> >
> > aeo6 Tony Orlow wrote:
> > > William Hughes said:
> > > >
> > > > Tony Orlow (aeo6) 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? 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.
> > > > >
> > > > > 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.
> > > > > I am not sure how to defined it to your satisfaction, but I think we all know
> > > > > what we are talking about. A finite number is one we could count to, and an
> > > > > infinite number is greater than any finite number.
> > > >
> > > > This is not quite what we need. We need a way to tell if a set has
> > > > a finite or infinite number of elements. We might use something like
> > > > "if we remove one element using a constant finte unit of time per
> > > > iteration,
> > > > we will always exhaust a set with a finite number of elements,
> > > > but never exhaust a set with an infinite number of elements".
> > > > Unfortunately, this leads immediately to the observation that any
> > > > set of integers with a finite number of elements has a largest element
> > > > (just take any integer from the set, then take the rest one by one,
> > > > always keeping the largest found so far. If the set has a finite
> > > > number
> > > > of elements this process must terminate. When it does you have your
> > > > largest element). So with this definition either:
> > > >
> > > > -there are an infinite number of finite integers
> > > >
> > > > or
> > > >
> > > > -there is a largest finite integer
> > > >
> > > >
> > > > > I think we agree that if x
> > > > > and y are finite, then x+y, x*y, x^y are all finite.
> > > >
> > > > The trouble is that "x,y finite implies x+y finite" leads immediately
> > > > to the fact that the sum of a finite number of integers is finite.
> > > > So:
> > > >
> > > > Let K be the set of finite integers. Assume K has a finite number of
> > > > elements. Let n be the sum of all the elements of K. Then n is a
> > > > finite integer. But n is not an element of K. Contradiction.
> > > > Therefore K has an infinite number of elements [1]
> > > >
> > > > -William Hughes
> > > >
> > > > [1] this specific argument was presented by Daryl McCullough
> > > >
> > > >
> > > Yes, I saw it. It's basically the "largest finite" argument.
> >
> > No it isn't. The difference is that you claim that:
> >
> > a: the largest finite natural does not exist
> >
> > b: the sum of all finite naturals does exist but cannot
> > be named.
> >
> > -William Hughes
> >
> >
> No, I never claimed the sum of all finite naturals "exists".

No I suppose you didn't. You have claimed that:

a: the sum of a finite number of finite integers is finite.

b: there are only a finite number of finite naturals.

I concluded from this that you were claiming that the sum
of all finite naturals exists. On the other hand I cannot see
how you can believe a and b and not believe that the sum of all finite
naturals exists.

>I proved that any
> set of strings up to a given finite length is finite, and cannot be infinite
> unless that length is allowed to be infinite. If all strings in your set are
> finite, then the condition required for the infinite set is not met, and as
> unbounded as your set may be, it is not infinite.
>

> The concept of summing all finite naturals rests on the concept of finding an
> end to the set at which to close the summation.

Maybe, but I am not making this argument. I am merely point out that
you have claimed that you can sum all the finite naturals (more
precisely
that you claim both a and b). Why you believe a and b (whether because
of some "end to the set" argument or for some other reason) is
beside the point. The fact that you believea and b is not.

> So, your argument is a thinly
> veiled largest-finite argument, no matter how you couch it.

My only claim is that we can use a and b because you have
claimed them. I did not make any representation as to why
you claimed them. Apparently you believe:

a: the sum of a finite number of finite integers is finite.

b: there are only a finite number of finite naturals.

c: a follows from the fact that there is an end to
the set of numbers you have to sum

d: there is no end the set of finite naturals so
you cannot sum them

Unfortunately, these do not form a consistent set. a and b imply that
the sum of the finite naturals exists, but d says this sum does
not exist.

-William Hughes

.

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