Re: Countable and uncountable infinities.

From: Dave Seaman (dseaman_at_no.such.host)
Date: 07/10/04 

Date: Sat, 10 Jul 2004 16:09:04 +0000 (UTC)

On Sun, 11 Jul 2004 01:35:50 +1000, Peter Webb wrote:

> "Michael N. Christoff" <mchristoff@sympatico.caREMOVETHIS> wrote in message
> news:PYQHc.42016$WM5.2239011@news20.bellglobal.com...
>>
>> "Kenneth Doyle" <nobody@notmail.com> wrote in message
>> news:Xns9522A72CD749Cnobodynotmailcom@61.9.191.5...
>> >
>> > Now, given an interval on the number line, there's an infinate
>> > number of points on the interval even though it's bounded.
>> >
>> > I'm hoping that the difference between those two ideas of
>> > infinity is something like the difference between a countable
>> > and an uncountable infinity. Any comments?
>> >
>>
>> The rational numbers are countable however there are an infinite number of
>> them in the bounded interval [0,1].

> I agree with all the above except for the last para, which talks about
> "rational numbers". While true, the OP was talking about real numbers, if I
> am not mistaken.

The OP asked whether the difference between infinity and unboundedness is
something like the difference between a countable and an uncountable
infinity. He asked for comments.

Michael Christoff correctly pointed out that they are not the same thing,
and he gave a counterexample. The rationals in [0,1] are bounded, but
infinite. Unlike the reals in [0,1], the rationals are countable. This
shows that uncountability is not an essential property of bounded
infinite sets. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

Relevant Pages

  • Re: Infinite Binary Strings: A Question
    ... counting is seen as the implementability of a bijection. ... But in the context of infinite sets of integers or reals, ... just as the rationals do, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > the rationals or reals here. ... and an infinite number of rationals for every unit. ... >>> substitute is mere handwaving and babbling. ...
    (sci.math)
  • Re: (sketch of a) Proof that the set of Real Numbers doesnt exist
    ... >> an infinite set and its powerset ... > both rationals and reals are numbers between rational numbers. ... assigned to rational numbers and pairs of open sets assigned to irrational ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... > Virgil wrote: Since the rest of WM's daydream is based on the ... > It is based on the existence of infinitely many rationals and on the ... > As it has not an infinite member, ... > naturals, then there are only elements, which count their initial sets. ...
    (sci.math)
  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... Among the notions of why there are "more" irrationals than rationals ... reals, ... Then, particular rationals of smaller numerators, after one and zero, ... as well in these infinite expansions have various ...
    (sci.math)
Loading