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Re: why do we distinguish between different kind of inifinity
Posted:
Jun 5, 2006 1:35 PM
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On 5 Jun 2006 05:40:08 -0700, flaxroot@yahoo.ca wrote:
> hi
> why do we need to distinguish between different kind of inifinity?
> (countable and uncountable).
> Is it being used to prove something important?
> please give a few examples.
> This is one of the questions I should have asked, when I was taking the
> course, but never did. I probably did prove something back then but
> have since forgotten all about it.
> Now it's bothering me again.
Here are several examples, some of which have already been mentioned.
It's possible for an infinite collection of positive numbers to have a
finite sum, but only if the collection is countable.
Every countable subset of the real numbers has measure zero. The
interval [0,1], which has measure 1, is therefore uncountable.
The Baire Category Theorem says that no complete metric space can be
expressed as a countable union of nowhere-dense sets. In particular, the
real numbers are a complete metric space, and it follows that the set of
real numbers is therefore uncountable.
Several useful subsets of the reals are known to be countable: the
rationals, the algebraic reals, the computable reals. From this we can
deduce that the irrationals, the transcendentals, and the noncomputable
reals are all uncountable and therefore nonempty.
A topological space is said to be "separable" if it has a countable dense
subset. In particular, the space R^n is separable for each n. Banach
spaces with dimension aleph_0 are also separable. Separable spaces have
some special properties, which merit giving them a special name. For
example, the continuous image of a separable space is separable. An open
subspace of a separable space is separable.
Besides separability, there are other properties of topological spaces
(first countability, second countability, countable compactness,
Lindelof, ...) whose definitions are intimately based on the concept of
countability.
> Is there no way around invoking such mind-bothering distinction?
The distinction is not meant to be "mind-bothering". It's meant to be a
useful classification criterion. Is it "mind-bothering" to distinguish
between finite and infinite sets? The countable/uncountable distinction
is equally important. A set is finite precisely when its cardinality is
< aleph_0. A set is countable precisely when its cardinality is <=
aleph_0.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
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