Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Topic: Is there a List of All Novels somewhere?
Replies: 30   Last Post: Jun 13, 2012 10:17 PM

 Messages: [ Previous | Next ]
 Guest
Re: Is there a List of All Novels somewhere?
Posted: Jun 1, 2012 8:21 AM

LudovicoVan wrote:
> "Aielyn" <gjo1984@gmail.com> wrote in message

>> This is just like how the rational numbers, which are countable
>> but infinite, can be listed according to the standard order,
>> and the diagonal doesn't prove it uncountable - why? Because
>> the cantor diagonal isn't a rational number - it's irrational,
>> and thus not missing from the list of *rational* numbers.

>
> How do you know that, while the diagonal argument applied to
> rationals provides a non-rational, the diagonal argument applied
> to reals does not provide a non-real?

It is a theorem of the axioms of real numbers, but not a theorem
of the axioms of rational numbers. The important difference is
that every bounded non-empty set of reals has a least bound.
This is not true of rationals.

We assign to every countably-long string of decimal digits
a,b,c,d,... of the form 0.abcd... the least upper bound of the
set { a/10, a/10 + b/100, a/10 + b/100 + c/1000, ... }.
We know this exists by the least-upper-bound axiom,
no matter which digits a,b,c,d,... are.

Apart from there not being a least-upper-bound axiom for
the rationals, we know that this is not true of them
by there being counter-examples, sets of rationals that
do not have a _rational_ least upper bound. This can
be seen from a very old theorem showing that there is no
rational square root of 2. The set { x in Q | x^2 =< 2 }
has a real, unique least upper bound, but not a rational
upper bound.

Date Subject Author
5/29/12 Graham Cooper
5/29/12 Pfsszxt@aol.com
5/29/12 Aielyn
5/31/12 LudovicoVan
5/31/12 Graham Cooper
6/2/12 Shmuel (Seymour J.) Metz
6/4/12 Aielyn
6/4/12 Graham Cooper
6/1/12 Guest
6/1/12 Graham Cooper
6/2/12 Jim Burns
6/2/12 David Bernier
6/2/12 Graham Cooper
6/2/12 Graham Cooper
6/3/12 David Bernier
6/3/12 Graham Cooper
6/6/12 David Bernier
6/6/12 Graham Cooper
6/6/12 David Bernier
6/6/12 David Bernier
6/9/12 David Bernier
6/9/12 David Bernier
6/9/12 �pam�uster
6/9/12 �pam�uster
6/13/12 Michael Press
6/13/12 Michael Press
6/13/12 Michael Press
5/29/12 Dimitry Yourdanov
5/29/12 Graham Cooper
5/29/12 Jesse F. Hughes
5/29/12 Graham Cooper