Rysto
June 28, 2009, 12:58am
1
An ongoing thread on Cantor reminded me of a book I own. It’s called “Infinity” by Lillian R. Lieber . I could describe it to you, but I think that you really have to read it to truly give it justice:

Having seen what
a paradox means(p. 316),
and having seen some paradoxes
in mathematics,
as well as
HOW THEY WERE ELIMINATED,
you will not be surprised to know
that paradoxes arose also in
the Theory of Transfinites,
and that
they too have been eliminated –
though of course
others may, and probably will,
arise in the future(see p. 314).
And you now know also that,
in spite of this uncertainty about
the future,
the “show must go on”
if we, AS A RACE
wish to LIVE and NOT DIE,
and that certainly
the story of mathematics is
THE SUCCESS STORY OF ALL TIME,
both in its THEORY
and in its
PRACTICAL APPLICATIONS

But first let us see how
Cantor showed that
c is NOT denumerable.
Since, as was shown above,
the LENGTH of the line-segment
has nothing to do with the case,
let us concentrate our attention on
the line-segment from 0 to 1,
which will therefore contain
the real numbers in this internal,
and each such number may be
represented by an infinite decimal,
thus:
0.e[sub]1[/sub]e[sub]2[/sub]. . . . .
where each e represents one of the
digits from 0 to 9 inclusive.

Now suppose that the set of
all real numbers in this internal,
from 0 to 1,
WERE “countable”;
we could then list them
and number them:
the first, the second, etc., etc.,
thus:

(1) 0.a[sub]1 1[/sub]a[sub]1 2[/sub] . . . . .
(2) 0.a[sub]2 1[/sub]a[sub]2 2[/sub] . . . . .
(3) . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .

this list being supposed to contain
ALL the real numbers in this interval.
BUT,
Cantor argued,
after considering this entire set as
being here enumerated,
you can now easily make up
ANOTHER infinite decimal which is
surely NOT in the list:
for you can make up a number which
DIFFERS from the first one in the list
in the FIRST decimal place
(namely, our new number
does NOT have a[sub]1 1[/sub] in its
first decimal place)
and DIFFERS from
the second number in the list
in the SECOND decimal place
(that is, our new number
does NOT have a[sub]2 2[/sub] in its
second decimal place),
and so on and so on,
being thus made up to
DIFFER from EVERY number in the list
by going down the diagonal and
changing the digit in
the decimal place you arrive at.

And so,
after thinking it IS possible to
ENUMERATE
ALL the real numbers in this interval,
from 0 to 1,
we find that
you can make up NEW ones
which have NOT been included in
the list!

Thus it is IMPOSSIBLE TO IMAGINE
a COMPLETE ENUMERABLE set of
real numbers in this interval,
or in any other interval.
Hence c is
a “transfinite number” which is
GREATER than Aleph-null! (see p. 111)

Yes, that’s right, it’s a book on mathematics as written by the Time Cube guy. Except that instead of being full of crackpot nonsense, it’s actually about wildly-accepted principles.

Incredibly, I actually learned something from this book when I first read it in the tenth grade(I had to quit half-way through when she started talking about “functions”). Despite the bizarre style, I actually found it quite readable.

Huh…I thought for sure this would be about this book I found in the kids section of Wal-Mart a few years ago.

Weston
June 28, 2009, 1:57am
3

Rysto:

An ongoing thread on Cantor reminded me of a book I own. It’s called “Infinity” by Lillian R. Lieber…

*…if we, AS A RACE
wish to LIVE and NOT DIE,
and that certainly
the story of mathematics is
THE SUCCESS STORY OF ALL TIME,
both in its THEORY
and in its
PRACTICAL APPLICATIONS. *

Lillian R. Lieber a.k.a. ‘Grandma Death’ from Donnie Darko, apparently…

It’s more of a physics book, but Milton W. Monson’s “Physics is Constipated” is an amazing and unintentionally hilarious book that points out how modern physics is all wrong, etc., etc.

And best of all, it’s WRITTEN IN ALL CAPS!

AMAPAC

Well, she had a good justification:

And her book on relativity is a lot more normal in terms of prose; if it weren’t for the layout or goofy illustrations, it’d almost seem sane.

Wildly-accepted? Is that anything like “wantonly-proven” or “heedlessly-substantiated”?

I just want you to know that I love this phrase and will use it at my earliest opportunity