OT: transfinite cardinals (was: Re: OT: White Goddess)
|From:||Dennis Paul Himes <dennis@...>|
|Date:||Friday, April 13, 2001, 4:17|
Yoon Ha Lee <yl112@...> wrote:
> On Thu, 12 Apr 2001, Bryan John Maloney wrote:
> > On Wed, 11 Apr 2001, Yoon Ha Lee wrote:
> > > Suppose you have a regular polygon with n sides. (I think you could
> > > get by with a weaker condition but this will suffice.) The "limit" of
> > > the polygon as n goes to infinity is a circle.
And, in fact, it is through this sort of limit that the length of a
nonpolygonal curve is defined. (See below.)
> > But it isn't a polygon with an infinite number of sides,
She didn't say it was.
> > because the
> > length of each of those sides would have, perforce, to be zero, which
> > means that the circumference, being a sum of zeros, would be zero.
The length of a curve is the limit of the lengths of line segment
approximations as the maximum length of a segment goes to zero.
> > As you say, the number of sides *approaches* infinity, but it is an
> > asymptotic limit. A curve is a curve, not a polygon,
A polygon is a curve, in the sense of the image of a continuous function
from a connected subset of R1 into R2.
> > but a curve can be
> > approximated by a polygon, if one actually wants to do something
> > practical. Approximation is not identity.
> Well, you *could* go the nonstandard analysis route. The length of each
> side would be infinitely small.
There's no need for that. A simple limit will suffice.
> If I have time later I can go into detail, or if someone
> else has the argument in memory (I would have to refresh mine) as to why
> Zeno was wrong (and why the above argument doesn't quite work), feel free.
Zeno makes unstated and untrue assumptions, such as "if a number of
events occur within a bounded period of time, then one of them must occur
first". That seems right intuitively, but is not in fact true.
The above argument makes its own intuitive but false assumption -- that
a sum of zeroes (using a very loose definition of "sum") must be zero. A
little thought will tell you that the union of sets all of measure zero can
have positive measure; else every set would measure zero.
> > Then say "transfinite", in that case. "Infinity" is not a number. It
> > is not a quantity. It should not be treated as if it were.
> Um--infinity isn't a real number, or a natural number, or a complex
> number. Nevertheless there *are* number systems in which various
> infinities are treated mathemtically as numbers. (Thank you, Cantor,
> even though the poor man died in an asylum, or at least spent a lot of
> time there.)
There are both infinite ordinals and infinite cardinals. And infinite
cardinals *are*, in fact, quantities. The number of integers is aleph
naught. The number of reals is c. You can do arithmetic on them, defined
such that when the same definitions are used to do arithmetic on the
natural numbers, you get the usual results.
In Gladilatian two transfinite cardinals have their own names:
srmo the cardinality of the integers
fryma the cardinality of the reals
The aleph and beth systems are formed by modifying _msoru_ "aleph" and
_msuto_ "beth" with ordinals. However, Gladilatian ordinals for the aleph
and beth notations are one off from English. So _zmrzno_msoru_, literally
"first aleph", is "aleph naught", and _zmrfsut_msoru_, literally "second
aleph", is "aleph one".
Dennis Paul Himes <> email@example.com
Gladilatian page: http://www.connix.com/~dennis/glad/lang.htm
"Zmrfsut msoru zmrfsut msuto" - the Continuum Hypothesis in Gladilatian