From: Jeremy Boden
Subject: Re: Question
Date: Tue, 23 Feb 1999 23:39:16 +0000
Newsgroups: sci.math
Keywords: What is the "Continuum Hypothesis"?
In article <7atv7l$k6s$1@usenet.kreonet.re.kr>, Pepero
writes
> Hello Everybody
>
> What is the "Continuum Hypothesis"?
> A brief conceptual explanation will be greatly appreciated.
>
> Thankyou
The Continuum Hypothesis(CH) is to do with measuring the "size", or
cardinality, of an infinite set.
The natural numbers, the integers and fractions all turn out to have the
same cardinality - aleph_0.
Perhaps surprisingly, the same is not true for the Real numbers which
have a larger cardinality, usually denoted by 'c'. The problem is what
cardinality should be allocated - i.e. which aleph?
One rendering of CH states that c = aleph_1.
It is known that CH is undecidable, within standard set theory.
A very readable description can be found at http://www.ii.com/math/ch/
--
Jeremy Boden mailto:jeremy@jboden.demon.co.uk
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From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: What is the deal with Cantor's Theorems?
Date: Fri, 19 Mar 1999 09:50:33 +1100
Newsgroups: sci.math
In article <36f16011.0@news.avana.net>, "Nick Tsocanos" wrote:
> I'm curious to understand why so many people think Cantor was wrong. I don't
> fully understand the proofs, but from what I've read they've yet to be
> proved as false. Could someone give me a small (laymen's) lecture on why
> people think it is wrong?
If A and B are finite sets and B has all the members of A and others besides,
then B is a bigger set than A. This is so deeply ingrained in us that there's
a natural tendency to think it should be true for infinite sets as well; I
think this is part of the reason for the resistance in some quarters to
Cantor's suggestion that the set of all integers is no bigger than the set
of even integers.
Then, there's an opposite tendency, to feel that infinity is infinity, and
that it makes no sense to say one infinity is bigger (more infinite?) than
another. Some people feel this very strongly & resist Cantor's demonstration
that the set of real numbers really is bigger than the set of integers.
So, Cantor gets it both coming and going.
I hasten to add that among mathematicians (as opposed to among people
posting to sci.math) there is no controversy here at all. Cantor's proofs
are fully accepted and no error is going to be found in them.
> I have read something about Continuum
> Hypothesis and I *think* Goedel proved it was an incomplete theorem? As in,
> it couldn't be proved or disproved?
Godel proved that CH is consistent with the other axioms of set theory;
it cannot be disproved (using those axioms). Cohen (1963) proved that the
negation of CH is also consistent with the other axioms, hence, CH cannot
be proved (using those axioms). The two results combined say that CH is
*independent* of the other axioms. It is the set of axioms that is
*incomplete*, meaning that there are independent statements like CH.
[Yes, I'm implicitly assuming that the axioms are consistent]
Gerry Myerson (gerry@mpce.mq.edu.au)
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["Cantor's Theorem" is the theorem that for every set X, the power set of
X is strictly larger than X itself; in particular, the real line is
not countable. --djr]
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