Re: before Cantor there was Rantor

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 10/17/04 

Date: Sun, 17 Oct 2004 16:00:06 GMT

In sci.logic, |-|erc
<spam@fodder.abc>
 wrote
on Sun, 17 Oct 2004 13:55:35 GMT
<rRucd.29652$5O5.2526@news-server.bigpond.net.au>:
> "The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in
>> In sci.logic, |-|erc
>> <spam@fodder.abc>
>> wrote
>> on Sat, 16 Oct 2004 23:58:38 GMT
>> <OAicd.28548$5O5.8103@news-server.bigpond.net.au>:
>> > "The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in
>> >> In sci.logic, |-|erc
>> >> <spam@fodder.abc>
>> >> wrote
>> >> on Fri, 15 Oct 2004 22:42:30 GMT
>> >> <qnYbd.27053$5O5.12770@news-server.bigpond.net.au>:
>> >> > "|-|erc" <spam@fodder.abc> wrote in
>> >> >
>> >> >> If you count an infinite list, there are numbers you
>> >> >> will never reach. That is a property of ordinary
>> >> >> countable infinity, not higher infinities.
>> >> >
>> >> >
>> >> > say you start with a white number line.
>> >> > you take your countable set of reals and mark in
>> >> > infinitesimally small black dot on that number.
>> >> > you cross of 10 numbers the 1st second, 100 numbers
>> >> > the next, then 1000, years go by and you're crossing
>> >> > off universes of avagadros of grahams numbers every microsecond.
>> >> >
>> >> > What color is the number line? its still white
>> >> > Does it matter if the set was countable or not?
>> >> >
>> >> > Herc
>> >> >
>> >>
>> >> The operation is better characterized in light of Rudin's
>> >> work on Lebesgue measure (the only work I'm even remotely
>> >> familiar with in this area), but the answer is that, after
>> >> taking the real line and painting it white (skirting Zeno's
>> >> Paradox if necessary), then painting all rational numbers
>> >> black, one would still have a white number line, as the set
>> >> of all rationals has measure 0 (under the standard metric).
>> >>
>> >> If one took an enumerated list of real numbers (1-1 isomorphism
>> >> with N), one would also get a white number line; all countable
>> >> subsets of reals have measure 0.
>> >>
>> >> Since the full set of reals does not have measure 0 it's clearly
>> >> not countable. Therefore, numbers exist outside of the countable
>> >> subset.
>> >>
>> >> One can also study Cantor's *first* proof, which proves an
>> >> identical result.
>> >>
>> >> What these numbers are, of course, this particular proof cannot
>> >> provide. Cantor's "diagonal" proof can provide one such number,
>> >> given a list; however, AIUI it's not rigorous enough -- mostly
>> >> because the actual decimal expansions specify a *range* of
>> >> numbers, as usually portrayed:
>> >>
>> >> 0.12378782... specifies some number a such that
>> >> 0.123787815 <= a < 0.123787825. We don't know which a,
>> >> although by convention, if the decimal is obviously
>> >> repeating, the standard assumption is that it is a
>> >> certain fraction that can easily be computed given the
>> >> existing digits.
>> >>
>> >> Since one cannot write an infinite list of numbers anyway
>> >> (the Universe dies first), one has to make some assumptions
>> >> when one sees the ellipsis. The most straightforward one is
>> >> a computable function D(i,d), mapping N x N to {0..9}.
>> >> Another computable function F(i) can map N x R. These
>> >> mappings are clearly not surjective (onto).
>> >>
>> >
>> > maybe youi're too old to learn new tricks. quoting dozens of
>> > mathematicians theorems that all tie in at some angle to
>> > hyperinfinities was the problem in the 1st place.
>> >
>> > just answer the question, is the sequence of anti-diag impossible?
>> > Herc
>>
>> Define "impossible". The anti-diag is perfectly computable,
>> given a computable list. In any event, I've answered your
>> "white number line" question.
>>
>> See also
>>
>> http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
>>
>> Did you have another question? :-)
>>
>
> say humanity evolved from the moment after Adam mated Eve,
> but Adam and Eve were folk just like you, you're actually
> a monkey with a primitive brain and have virtually no
> comprehension of the world around you. the above average
> apes have enough iq to realise a paradox with numbers but
> are too dim to realise what it actually signafies. along
> comes Adam who is superpowered by a universe sized quantum
> field, he can't throw a dart over his shoulder without it
> hitting the bulls eye, alas he is the only one who can see
> that anti-diag is an illusion.
>
> could the apes one day learn what random genetic animal freaks
> they really are?

Is *this* your next question?

I do not have an answer therefor, as you are postulating
a hypothetical in the future, with insufficient data -- assuming
that the implied subquestions in your question:

[1] that we are primitive apes
[2] that we are genetic animal freaks
[3] that we are capable of learning that we are primitive apes
    and genetic animal freaks

actually hold true. We are not primitive apes; we are
hominids, though still rather primitive if one takes
certain sci-fi notions into account -- one particularly
grisly possibility, for example, is cutting off one's
head and then immediately transplanting it into a robot
recirculatory unit, creating a cyborg (one can think,
perhaps, of _The Ultimate Enemy_, by Fred Saberhagen; it's
a collection of stories relating to various victories and
defeats in a universe with berserkers, which are trying to
kill all life -- but in order to more successfully kill,
they conduct rather sadistic experiments with human brain,
tissue, etc.); another, slightly milder possibility
involves encasing the brain in a large fluid-filled
cavity, then removing the skull and allowing the neurons
to float and be fed by the liquid, creating a gigantic
brain much larger in size to control a spacecraft (this
is a combination of factors; the closest coming to mind
is an episode of _Farscape_). Other possibilities include
reading the brain using X-rays or NMI and then recreating
the neuron hookups, duplicating (one hopes) the intelligence
represented therein; still others involve classical AI
concepts, recreating intelligence such as the HAL 9000 or
Isaac Asimov's world-rulers (cf his _I, Robot_ anthology;
I forget the precise story therein). So far, we've been
able to best chestmasters by brute force, but computers
remain fairly stupid.

I'm not sure what a "genetic animal freak" is, in
this context. We are what we are. The Nephilim remain
unproven, though I've not read _12th Planet_ (or is it
_Twelfth Planet_?); apparently the main hypothesis therein
is that we are genetic creations thereof, and that the
planet is describing a rather elliptical orbit with a
period of 26,000 or so years, and that these superbeings
last visited either 2,000 or 6,000 years ago, I forget
which. Other possibilities exist regarding our creation,
if one is sufficiently creative in interpreting how we
are created from dust, mud, or clay in the Bible. We
are at best created from meat and vegetables ("you are
what you eat"), which come from soil, but that's not
really a sufficient answer since most animals are created
from meat (carnivores) or vegetables (herbivores, omnivores).
The only real difference between humans and other animals
is our body/brain mass ratio, and the dolphins give us
a run for our money there, and show very intelligent
behavior in catching fish, for example.

And of course most of us are capable of learning.
(Feel free to interpret this as you will.)

I am also wondering if Adam and Eve -- or, to be less
religious about it, an arbitrary pair of human beings
(male and female) around 6,000 years ago -- are actually
the same as us, and how to characterize the difference.
There is a concept called "genetic drift"; the next
generation of a species will be different from the current
one because of this drift. I've no data on the actual
amounts of the drift, and characterizing it is difficult
if not impossible without a fair notion of how one can
determine a "genetic map". The Human Genome Project is
AFAIK a catalogue of genes, but how does one determine the
distance between blue eyes and brown, between the presence
of melanin and the absence of one's sensibility to pain
(a defect present in a very few individuals which causes no
end of trouble, as they can't sense damage to themselves),
between the ability to curl one's tongue in a U-shape and
the predisposition towards arthritis, Hodgkin's disease,
Lou Gehrig's disease, glaucoma, or diabetes?

Also, as someone once said, "clothes make the man".
This someone is slightly but not far off; I'd say in this
case *tools* make the man, and it's a superregenerative
process; better tools make for better results which makes
for more sophisticated and better tools which makes for
better results. Contrast today's PC with units in the
1946 era, of which there might have been two or so (ENIAC
is the only one I know of, but presumably the Soviets of
that era were busy researching computers at some point),
or Sputnik with today's communications satellites, or even
the light bulbs of today with the antiquated units that
might still be burning in some firehouses.

Of course clothing is better today than it was hundreds of
years ago, although the only specific example I can think
of offhand is rayon.

Ultimately it could get *really* weird, though; cf.
Stanislaw Lem's 21st voyage in his _Starship Diaries_
for one rather fanciful possibility. (One connection
to reality, which is more disturbing, is that an
attempt to splice in an eye expression gene in a fruit
fly lead to rather unexpected results; they wanted an
eye on the knee, IIRC, but got eyes all over, instead.
Must have been painful for the poor fly, if flies can
sense pain of any sort.)

You are also on record -- I'd have to find the specific post(s)
admittedly -- as stating that you are being mind-controlled
via satellite. I suspect you're rather ignorant of the brain,
which would be rather difficult to control (except through
rather gross means, e.g., drugs) in any event. The best
one might do in this area would be to surgically remove the
back of the head and insert a probe-array into the visual
cortex; the only research I know of being done in this area
has led to promising but rather low-fidelity results, mostly
because the probe-array was all of 25 units. However, the
individual after this operation could distinguish light
from dark, which is an improvement.

We still don't really know what a thought *is*. (At least,
not that I know of.) Of course, we have some ideas,
and with modern computers we might have a chance at
simulations to check our results, given enough data.
I'm of the opinion that a thought is a bit like a hologram.
Douglas Hofstadter in his work _Gödel, Escher, Bach: An
Eternal Golden Braid_ goes the other way, hypothesizing a
"grandmother neuron" (which fires whenever other neurons
feed it signals consistent with the observation that there
is a grandmother nearby). Both may be true; I don't know
at this point. Both brain research and neural networks (a
simulation of thought in modern computers) are continuing.

But I ramble. In short, you are fighting the Establishment.
While the Establishment can be fought -- cf the more or less
interesting situation of Einstein's theory of special
relativity published sometime in the early 1900's (1905?
I'd have to look) -- it's clear that Einstein had a lot
more going for him than you do:

[1] photoelectric effect
[2] anomalies of Mercury's orbit
[3] MMX null result

if memory serves. (Other results came later, and indeed
modern accelerators have to take into account the predicted
SR mass-gain, lest they provide undesired results; GPS
satellites also have SR and GR compensations built-in.)

Of course, you're battling the hellions of pure
mathematics, not a combination of math and physics (which
admittedly get more intertangled every year). Your job
is harder; mathematics, after all, is pure abstractedness.
One cannot construct a real number line; the best one can
do is scratch some carbon down on a piece of paper with a
standard #2 pencil, or dribble some ink (tannic acid, IIRC,
though there are many forms of ink) using a ball-point
pen -- itself an interesting product of technology,
as the metal ball and the ink have to work together --
or melt some plastic particles on a piece of paper using
a laser printer, or hit the paper with a hammer through
an inked ribbon, or even chisel hardened stone or use a
pointed tool in clay, or older-model quills needing
periodic sharpening and then dipped in ink.

And numbers can't be caught in a butterfly net, although
one might try Erastothene's sieve to catch some of them... :-)

Or, as you and I are doing here, one can argue through
the movements of electrons, laser pits (CD burning),
and magnetic domain manipulations as the head flies at
high speed and very small distances over a mirror-smooth
spinning platter somewhere deep inside a sealed unit, which
is itself usually sealed inside of a (usually) gray or
black box, either standing on a desk or attached to a rack.

I surmise the reader is a better judge than I on how your
arguments are going. I'd say they're not going at all well.
For starters, one has to battle the following.

[1] Cantor's second proof, based on the "anti-diagonal" notion.
    While the proof has some minor problems (at least one
    paper attempts to fix them) it's held up rather well.

[2] Cantor's *first* proof, which is described in detail in

http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof

    and requires no decimal expansions at all.

[3] Rudin's work on Lebesgue measures, and other such issues
    (a countable set has measure 0).

You may think you've defeated [1], but [2] and [3] are laughing at you.
And no,

http://zapatopi.net/afdb.html

will not help you, though Elizabeth Hurley is pleasantly distracting... :-)

>
> Herc
> 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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