[thechat] Cantor Diagonal - was Math was Religion was Life, the Universe an d Everything
Luther, Ron
Ron.Luther at COMPAQ.com
Thu May 24 16:11:57 CDT 2001
Hi Andrew,
Naw ... it's not a 'Barber of Seville' paradox or anything ... it's a proof
by a really strange guy (Gregor Cantor) that the number of Real numbers in
any given interval on a number line is strictly larger than the number of
integers. {A proof that there are an uncountably infinite number of Real
numbers as opposed to a countably infinite number of integers.}
Note 1 - He was a strange guy. If I remember rightly his musings on
different sized infinities ended up with him loosing his sanity. I believe
the story went that a student was hired to play chess with him all day every
day to help him regain his sanity. Story says it worked ... but the student
went insane!
Note 2 - I mistakenly referred to countably infinite as "aleph-not". Doh!
That should have been "aleph-naught"! [It looks like a capital German
script "X" with a subscript of zero.]
... pulling the Cantor diagonal argument from memory - detail may not be
perfect but the crux should be correct ...
Assume that the Real numbers in the closed interval from zero to one [0, 1]
are "countably infinite".
If they are "countably infinite" then they are enumerable.
Enumerate them:
.1010101918116161661616
.10202378934729387492387
.28912734979823479
.323874923874289374923
etc.
Okay.
Now take the number at the top of your list. Change the first digit to the
right of the decimal. Take the second number on your list - change the
second digit in that number. Take the third number on the list - change the
third digit in the number. ... and so on ...
The 'new' number you have created in this 'diagonal' manner is not in the
original enumeration. [It differs (by construction) from any number in the
original list in at least one position.]
Therefore your premise that the real numbers in that interval can be
enumerated is flawed.
Hence the number of real numbers in the closed interval between zero and one
in uncountably infinite.
If you take enough math, you'll see a lot of neat ideas (like this one) get
"recycled" over and over again in slightly different applications. "New"
ideas are pretty darn rare. If I remember correctly [and I'm really going
to have to go and check] Godel's theorem was a blockbuster ... but the proof
was kind of a rehash of the above diagonal argument in a new context.
Where will I check? There is a 4 volume "World of Mathematics" edited by
Neuman (sp?). It's a fabulous resource for the 'extreme math geek'. Want
to learn about relativity? Fine, the original Einstein paper is in there.
Want to learn about logic? Fine, original papers from great logicians are in
there. etc. etc. There's also Doug Hoffsteader's (sp?) very nice "Godel,
Escher, and Bach" book from a few years back.
This 'math moment' brought to you by,
RonL.
... for best results, pronounced, "Okay Ron, here is my signed blank check"!
-----Original Message-----
From: Andrew Forsberg [mailto:andrew at thepander.co.nz]
Subject: RE: [thechat] Math was Mythology was comics
Is the Cantor diagonal argument anything like (he asks hopefully) the
Liar's paradox or Protagoras's argument with his pupil:
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