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Re: Is there a List of All Novels somewhere?
Posted:
Jun 1, 2012 8:21 AM
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LudovicoVan wrote:
> "Aielyn" <gjo1984@gmail.com> wrote in message
> news:e01347e5-cf03-452b-8863-6d2121e6afae@googlegroups.com...
>> 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.
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