Natural Numbers

Natural numbers, denoted by the symbol ℕ, are the numbers used to count. As such, they represent the most immediate connection that a mathematical abstraction, the number, has with the objects of the physical world

They are used to count, but do they start from 1 or 0?

Precisely because of their close connection with material things that can be counted (goods, animals, people, etc.), some mathematicians hold that the series of natural numbers must begin with 1. Others think, however, that it must start with 0, partly because this choice allows a rigorous formalization of the concept of natural numbers, such as that provided by the so-called Peano axioms.

Because of this ambiguity, the symbol used to denote the set of natural numbers is often accompanied by a clarifying character in subscript or superscript:

• ℕ₀ indicates that the series starts from 0; in this case, the set of natural numbers corresponds to the set of non-negative integers (i.e., all positive numbers plus 0);
• ℕ₁, ℕ₊, ℕ* indicate that 0 is excluded and that the series starts from 1; in this case, the natural numbers correspond to the set of positive integers.