Re: Can You Have Infinity Without Eternity?
- From: Ittay Weiss <ettaybn@xxxxxxxxx>
- Date: Fri, 09 Sep 2005 06:15:21 EDT
>
> "Schoenfeld" <schoenfeld1@xxxxxxxxx> wrote in message
>
> news:1126230733.834081.234980@xxxxxxxxxxxxxxxxxxxxxxxx
> .com...
> >
> > DBLEXPOSURE wrote:
> >> "Sleepyhead" <simonharpham@xxxxxxx> wrote in
> message
> >>
> news:1126202185.982669.166390@xxxxxxxxxxxxxxxxxxxxxxxx
> .com...
> >> > Yes, but only if the word "Infinite" is taken to
> mean "A number too
> >> > large to comprehend". If you simply never stop
> counting it doesn't make
> >> > any difference which set of numbers you're
> counting because
> >> > (ex-hypothesi) you'll never reach the end.
> >> >
> >>
> >>
> >> Alright. I am stepping into a realm wear I am
> surely to be bludgeoned to
> >> death. I am neither a physicists nor a
> mathematician, I just find it
> >> all
> >> very interesting.
> >
> > For almost all purposes, infinity can be considered
> a 'dead ender' in
> > your words. But more precisely, there are infinite
> sets with
> > cardinality (size of the set) greater than others.
> >
> > Consider two infinite sets A and B:
> > 1. If a surjective injective mapping (a bijection)
> exists between A
> > and B
> > then |A| = |B|
> >
> > 2. If a non-surjective injective mapping exists
> between A and B
> > then |A| < |B|
> >
> > 3. If a non-surjective injective mapping exists
> between B and A then
> > |A| > |B|
> >
> >
> > For example, the infinite sets A=[0,1] and B=[-inf,
> +inf] have equal
> > size since a bijection f exists between the two
> sets.
> >
> > Example:
> > f(x) = tan( pi x - pi/2 ) maps [0,1] onto [-inf,
> +inf]
> >
> >
> >
> >> Is not Infinity a mathematical dead ender? You
> cannot apply any
> >> mathematical
> >> operator to it. It can't be divided, multiplied,
> added to or subtracted
> >> from. Doesn't this tell us that either it is
> impossible or that our
> >> understanding of mathematics is flawed. Perhaps
> the universe is not
> >> Infinite. I can better understand absolute
> nothing better than, or more
> >> easily than infinity or a never ending universe.
> Could it not be that
> >> the
> >> universe slowly fades away to a point where
> matter cannot exist. That
> >> space time itself is a gradient and behaves
> differently at it's edges
> >> than
> >> it does at the center? To me this seems more
> comprehendible...??
> >
> > If you read the following, it may become more
> comprehendible. The
> > cardinality of the integers |Z| is the first
> infinity (aleph\0). If you
> > can construct a bijection between the set of
> integers Z and another
> > infinite set A then |A| = |Z|. The rationals are an
> example of |A|.
> > Since one cannot construct a bijection between the
> set Z and powerset
> > of Z, the cardinality of powerset of Z is greater
> than |Z|. The
> > cardinality of the powerset of Z is 2^|Z|, and is
> considered the next
> > logical infinity aleph\1. More precisely:
> > 2^|Z| = aleph\1
> >
> > Since one can construct a bijection between the
> reals R and powerset of
> > Z, the cardinality of the R is aleph\1. These
> infinities are called
> > transfinite numbers. There are believed to be
> aleph\0 transfinite
> > numbers. It was hypothesized by the great
> mathematican Georg Cantor
> > that for all transfinite aleph\k:
> > aleph\(k+1) = 2^(aleph\k)
> >
> > This is known as the Continuum hypothesis and
> remains unproven to this
> > day. The Continuum hypothesis implies that no set
> with cardinality
> > between aleph\k and aleph\(k+1) exists. Godel
> proved this hypothesis
> > undecidable within Zeremelo-Frankel Set theory.
> Godel also showed that
> > no intrinsic contradictions arise in mathematics if
> this hypothesis was
> > false, not so if it were true.
> >
>
> Lol.. I think I'll stick to philosophy... I
> appreciate you trying to, dumb
> it down, for me but that was a bit like reading
> glyphs... My eyes are stuck
> in a cross eyed position.
>
> Even so, I think I understood enough to ask a
> question. Does any of what
> you just, so eloquently, said have any application in
> reality? Is there a
> real example where this math would apply?
In ordinary physics people tend to use R^3 as a model for the world around us. This seems to be quite a good model. Notice that the size of R^3 is infinite. Now nobody is saying that R^3 is an exact model of the real world, but the fact is that using this infinite model physicists come up with lots of good stuff!
>
> The, "Finite", amount of gray matter that exists
> between my ears is telling
> me that the only real thing we have that may approach
> infinity is our own
> universe. And if you might entertain my silly idea
> that the universe is a
> gradient. That is to say that space time slowly
> fades away to a point where
> there is not enough energy to form matter. If this
> where true the universe
> would,(could) then be a known quantity and then so
> would infinity,(re
> writing the definition). Perhaps then Godel's
> equation could be proved?
>
what is Goedel's equation?
> Also, if this where true wouldn't it be hinting that
> perhaps unification
> exists at the outer edges of our universe :-) Life
> is great when you
> don't have to back it up with numbers......
>
> As always, please excuse my ignorance...
>
>
>
>
>
>
.
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