Does Aleph-null Share Indeterminate Qualities with Infinity?

Date: 10/19/2004 at 13:11:15
From: Leo
Subject: Is aleph-null associated with indeterminate expressions?

I know that the following expressions are indeterminate (where I 
represents infinity): I*0, I^0, 1^I, I/I, and I-I.

I know that aleph-null is the cardinal number of the set of counting 
numbers.  But conceptually I find it difficult to dissociate the 
concept of aleph-null from infinity.

If I use a set N to represent the cardinality of an infinite set, then 
I imagine that N takes on the attributes of infinity.  Thus, I wonder 
whether it is correct to associate the indeterminate qualities of 
infinity with aleph-null?

Date: 10/20/2004 at 15:48:45
From: Doctor Schwa
Subject: Re: Is aleph-null associated with indeterminate expressions?

Hi Leo,

I would say that "infinity" as a "number" in the sense of those
expressions you get from taking limits of things is QUITE different
from "infinity" as the cardinality of a set.

For instance, to define "infinity * 0" as a limit, you'd be saying
something like, "What happens to f(x) * g(x) as x gets large, if f(x)
keeps getting bigger without bound and g(x) keeps getting closer to
0?"  When you say that, then the answer depends on exactly how fast
f(x) gets bigger and how fast g(x) gets close to 0. (I've paraphrased
a bit--this is not good enough to be the mathematical definition, but 
it gets the idea across, I hope).

When you talk about aleph_null, you're talking about how many items
are in a set.  Then, "+" is defined as "number of things in the union
of two disjoint sets with that cardinality"--for instance, 1 + 2 = 3
because if you take {apple} union {bear, cat} you get {apple, bear,
cat} which has three things in it.

Multiplication, more relevant for your question, would be the number 
of things in the set of ordered pairs, where the first element of the 
pair is from the first set and the second element is from the second 
set.  For instance, 2*3 = 6 because if you take {a, b} and {1,2,3} you 
get {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)} which has six things.

Now, from that definition, it's pretty obvious that aleph_null * 0 =
0, because if there's nothing in the second set, then you can't make
any ordered pairs!

So, for these reasons (among many others!) the concept of infinity
in indeterminate forms of limits is quite different than the concept
in cardinality of sets.

Some of the many others might include things like; infinity is not a
number, but aleph_null actually IS the cardinality of a set (such as
the set of natural numbers).

I hope that helps clear things up!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/22/2004 at 15:12:01
From: Leo
Subject: Thank you (Is aleph-null associated with indeterminate
expressions?)

Dr. Math--

Thank you so much for your quick response.  It helps me considerably 
to consider the context in which infinity is examined, e.g., as a 
limit or as cardinality for an infinite set.

-Leo