Aleph Null

Date: 01/22/98 at 19:08:14
From: Jonathan
Subject: Aleph null

What does aleph null represent?

Date: 01/27/98 at 16:19:34
From: Doctor Joe
Subject: Re: Aleph null

Dear Jonathan,

Before I deal with Aleph Null formally, I need to go through some 
stuff about sets, first the finite ones, then the infinite ones; and 
last of all I shall explain how you measure the size of these sets.

A set is a collection of objects. An object is called an element.

For instance, a class of K students is regarded as a set of students.

Some sets are finite (meaning there is a finite number of elements in 
those sets) while some are infinite (for instance the set of all 
integers).

If a class of K students has 12 members(or elements), then K has size 
12. Technically, we say that the set has order 12 or we say that the 
cardinality of the set is 12.

For infinite sets, we are also interested in whether we can somehow 
count the objects. Of course, we can never finish listing them. But 
what we may do (as best we can) is count the elements by assigning the 
natural numbers in a one-to-one way, so as not to skip any members of 
the set.

This concept leads us to the formulation of countability.

A set is said to be countable if either it is finite or it can be put 
on a one-to-one correspondence with the set of natural numbers 
{1,2,3,...}.

For instance, we say that the set of integers is countable since 
the following listing demonstrates that there is a one-to-one 
correspondence between the set of integers and the set of natural 
numbers:

    0  1  -1  2  -2  3  -3 ...
    |  |   |  |   |  |   |
    1  2   3  4   5  6   7 ...

If a set X has a one-to-one correspondence with the set of natural 
numbers N, then X is said to have the same cardinality as N. 
And the cardinality of N is denoted by the first cardinal number 
Aleph Null.

-Doctor Joe,  The Math Forum
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