Re: CHAT: mathematics

On Mon, Nov 20, 2000 at 10:21:49PM -0500, Dennis Paul Himes wrote:
> Carlos Thompson <carlos_thompson@...> wrote:
[snip]
> meaning of "complete" in mathematics).  The reals, in fact, can be defined
> in terms of equivalence classes of Cauchy sequences of rationals.  Because of
Awesome! So *that's* what Cauchy sequences are all about. And I was
staring at my number theory book for the longest time, and couldn't make
head or tail of Cauchy sequences. (Or those infinitely-nested fractions,
for that matter. All I remembered was some fantastic claim about how
those fractions sometimes converge to irrationals...)

> this, we say that the reals are the "completion of the rationals".
>
> > This means that Q is full of wholes.
>
>     That's "holes".
Hmm. This could be a punny fun, too. Um, I mean, a funny pun. :-) Q *is*
full of "wholes" -- whole numbers, as well as "holes", the missing
irrationals. [Tangential random thought:] In fact, you could just as well
say that Q is almost completely filled with holes, with an occasional
rational or two, because the cardinality of the irrationals is much larger
than the cardinality of Q (infinitely larger, to abuse the term a bit)!
So, rationals are actually mere needles in an infinitely large haystack...
go figure. :-P


T

--
Caffeine underflow. Brain dumped.