"WM" wrote in message news:email@example.com... > > It is not a natural number and it cannot count the number > of natural numbers.
Depends on how the word `count' is defined. The natural numbers are said to be countably infinite. There is no natural number that counts the total number of natural numbers so a new type of number is needed to "count" them. The ordinal w is used to define aleph_o but whatever number you want to use to "count" natural numbers it must be infinite.
> Every set of natural numbers has less than aleph_0 elements.
The set of odd natural numbers has aleph_o elements, for example.