Re: No Unique Initial Segment And No Characteristic Expansion.
From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 12/03/04
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Date: 3 Dec 2004 14:21:51 -0800
Uncle Al <UncleAl0@hate.spam.net> wrote in message news:<41B0AC7A.A71D917B@hate.spam.net>...
> HERC777 wrote:
> >
> > Infinite people each flip coins infinite times.
> > Can you always find a different sequence of heads and tails?
>
> Wait for the data to fuly accumulate, then look. Idiot. Given n
> flips there will be 2^n states. That will nicely outrace any
> approximation to infinity you attempt.
>
> > sci.math and sci.logic went quiet on this question for about a week,
> > then against all logic, probability theory, and common sense they all
> > agreed YES. Believers of hyperinfinities have no shame!
>
> Georg Cantor. As for you, FOAD.
>
> [snip crap]
Hi Al,
Hey, for a length n, you get 2^n permutations, and they're all
rational numbers. In base b, for p places, you get b^p permutations.
You're a chemist, and respected for your knowledge of chemistry. Some
people think you're great, others you're an over-the-top jerk, but
it's generally accepted that you're a reliable scientist. I don't
care about "Herc", it's a free country. So anyways, I want to know if
you ever use the uncountability of the reals or transfinite
cardinality in any form.
Georg Cantor's famous. He might be the most chronicled mathematician
after Newton, and I never heard of him until the late 90's. He
tackled the difficult task of trying to define the continuum in
non-geometric terms.
During the rush to reformalize mathematics in the 20th century,
basically the Bourbaki school and perhaps Dresden with Russell and
Whitehead being Englishmen readily used this notion of "set theory",
and along with it Cantor's powerset mapping result. As the century
progressed, in terms of foundations there are basically the Zermelo
and Fraenkel, and the Goedelian period and the 70's, and mathematics
at large flourished outside of it, with tremendous gains in concrete
and differential mathematics since the 50's and 60's, and of course
the tremendous jumps in the tensorial and algebraic geometrical and
geometric algebraic methods, and of course the probability density
functions in your field, and the unfortunately ready application of
numerical methods with digital computers.
By and large those have little to do with foundations. Anyways, today
there are theories that do not necessarily use ZF, for example, an
excellent set theory with a somewhat restricted notion of a set, and
calling everything else a class, except the class can not contain
other classes, leading to no class of all classes, and no reolution of
Burali-Forti. There are a variety of rigorous infinitesimals these
days, some hundreds of years after the fact of their use for results.
I retract that calling something a paradox is stupid, but stopping
there is stupid. A paradox is a sign of insufficient knowledge or
improper assumptions.
About the semi-infinite binary sequences, they are that, but they
might not adequately characterize the real numbers or the continuum.
Today, there are a variety of modern approaches to foundations that
attempt to revisit the assumptions and expand the knowledge about the
behavior of fundamental mathematical objects.
Some of these do not have the powerset result holding for variously
many or all sets in the theory.
One of them is mine, currently a theory with ubiquitous ordinals, and
the ur-element, or that which is assumed, is either a) empty, or b)
the universal set. Also it doesn't say.
Basically its strength is to get over Burali-Forti, that the order
type of all ordinals would itself be an ordinal, which is the same
thing as Cantor, that the set of all sets would be its own powerset.
So anyways, do you use transfinite cardinals in your day-to-day
operations or ever?
What's the use of integrating over the natural numbers and getting a
meaningful result?
Fondly,
Frere Ross
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