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Re: Problems with Infinity?
Posted:
Feb 25, 2013 1:23 AM
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On Feb 24, 2:01 pm, Quadibloc <jsav...@ecn.ab.ca> wrote:
> On Feb 24, 12:00 am, Don Kuenz <garb...@crcomp.net> wrote:
>
> > Let's say a Mobius strip goes to infinity "feedback style" (in layman's
> > terms) while a line goes to two separate but equal infinities "linear
> > style." How many different infinities does that make according to
> > Cantor? One, two, or three?
>
> According to Cantor, the number of points on a line, or on a circle,
> has the cardinality of the continuum.
>
> The length of a line is aleph-null finite units of measure, on the
> other hand - and the length of a Mobius strip is finite, the distance
> it takes for you to get back where you started.
>
> Cantor's infinities, as was noted, aren't really about things like
> that.
>
> Basically, aleph-null is the first kind of infinity - the number of
> integers in the set {1, 2, 3, 4, 5, 6... }.
>
> It can be proven that one can pair off all the integers with that set:
>
> 1 <-> 0
> 2 <-> -1
> 3 <-> 1
> 4 <-> -2
> ...
>
> and so including the negative numbers doesn't really "double" that
> infinity. In fact, even the rational numbers can be paired off with
> the integers.
>
> Cantor's diagonal proof shows, though, that there are more *real
> numbers* than integers in a very real and unavoidable sense.
>
> http://www.quadibloc.com/math/infint.htm
>
> discusses the subject in more detail than I can do in a post.
>
> John Savard
Discussed the subject in somewhat more detail than a single post.
Arrange real numbers in a line this way. For naturals n from 0 to d,
arrange n/d, in the order of the naturals. As d (simply enough for
denominator) diverges to infinity, there are arrayed points between
zero and one, with a constant, infinitesimal difference from one to
the next, uniformly between zero and one.
The corresponding antidiagonal is at the end of the list, and nesting
intervals finds there's just the one for (f(0), f(1)).
Then, the antidiagonal argument doesn't apply to this function, nor
does the nested intervals argument. Then with regards to set-
theoretic machinery there is a reasonable consideration in ordinals.
The Universe, as a set, would be its very own powerset, equivalent via
equality. A variety of modern cosmological theories have that it is,
containing itself in a real way. Speaking of real, re the reals,
there aren't known uses of transfinite cardinals in physics. And, it
is so that physics is continuum analysis, with infinities, and needs
mathematics of the infinite to best explain some things. The
multitudes of cardinals aren't ignored, but mute, in the magnitudes of
measure, with the continuum divided, measurably, countably. So,
besides finding examples of the modernly infinite, in the mathematics,
in fiction, find some in reality.
Infinity makes natural integers.
Train rolls on. So: draw a line.
Regards,
Ross Finlayson
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