Re: abundance of irrationals
From: J (Jaybirdmac_at_yahoo.com)
Date: 01/08/05
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Date: 7 Jan 2005 18:01:25 -0800
Your proof basically says that if given two infinite sets, take a
member of the first infinite set and pair it up with a member of the
second infinite set, then take a different member of the first infinite
set and pair it up with a different member of the second set and
continue in this fashion forever. You have shown that both sets are
infinite. Nobody is disagreeing with that.
Cantor found a way to list the rational numbers in such a way that no
rational number would be overlooked. Even though the list of rational
numbers is infinite, mathematicians agreed that the list he devised was
sufficient to accomplish this denumerability. Denumerable doesn't mean
infinite, it means countable just as the natural numbers are countable.
You can count the natural numbers forever and never reach the end, but
you can be assured that you won't miss a member of the natural numbers
if you start with one and add another one to get two and add another
one to get three, etc.
Cantors list of rational numbers went something like this: 0/1, +/-
1/1, +/- 1/2, +/- 2/1, +/- 1/3, +/- 3/1, +/- 2/3, +/- 3/2, ... where
the ellipsis means goes on forever and based on the unambiguous
definition of the rational numbers being a ratio of two integers as
long as zero is not included in the denominator of one of the ratios.
This method of listing the rational numbers assures that no rational
number is missing, overlooked or skipped, even though the set of
rational numbers is infinite. Notice the list of rational numbers that
Cantor devised is not in order from left to right on the number line
like the list of natural numbers. Nevertheless both the natural numbers
and rational numbers are denumerable and so belong to the same class of
infinite sets.
The set of irrationals even though infinite are not denumerable. If you
want to learn more about this, check the Mathematical Association of
America web site, Maa.org, and order some of their books in The Carus
Mathematical Monographs.
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