Notation
Mathematicians use N or (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null .
To be unambiguous about whether zero is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "" or subscript "" is added in the latter case:
Some authors who exclude zero from the naturals may distinguish the set of nonnegative integers by referring to the latter as the natural numbers with zero, whole numbers, or counting numbers, denoted W. Others use the notation P for the positive integers if there is no danger of confusing this with the prime numbers. In that case, a popular notation is to use a script P for positive integers (which extends to using script N for negative integers, and script Z for zero).
Set theorists often denote the set of all natural numbers including zero by a lower-case Greek letter omega: ω. This stems from the identification of an ordinal number with the set of ordinals that are smaller. Moreover, adopting the von Neumann definition of ordinals and defining cardinal numbers as minimal ordinals among those with same cardinality leads to the identity .
Read more about this topic: Natural Number