In a sense hodgepodge
is correct, and in a sense he is not. It would perhaps be more correct to say that a set with cardinality
aleph-null is the smallest set such that there is no natural number to represent the number of elements in the set.
A set with cardinality aleph-null is a set which can be put in one-to-one correspondence with the set of natural numbers. These sets are called "countable" sets for this reason. Many important sets in mathematics are countable, such as:
One useful property of a countable set is the ability to put a non-dense ordering on it. This means that one can define "less than" such that there is no number "between" two adjacent numbers. This is very useful for certain kinds of proof, e.g. proof by induction.
As with all the transfinite numbers, aleph-null does not obey the same laws of algebra as finite numbers. Here are a few basic facts about algebra involving aleph-null:
k + aleph-null = aleph-null for any finite k
k aleph-null = aleph-null for any finite non-zero k
aleph-nullk = aleph-null for any positive k
However, kaleph-null = aleph-one for any finite k > 1
Yes, I know a lot of this is a bit wrong. I've ignored lots of horrible stuff about the Axiom Of Choice. This just gives a basic overview of the concept. If you want more grisly details, read Gorgonzola's write-up under cardinal.
This has been part of the Maths for the masses project