Natural numbers (starting at 0)
Elementary
The natural numbers are the numbers $0, 1, 2, 3, \dots$ that can be used to answer questions about “how many?” objects there are.
Undergraduate
The natural numbers are the set $\lbrace 0, 1, 2, 3, \dots \rbrace$ of nonnegative integers, often written $\mathbb{N}$, \(\mathbb{N}_0\), or \(\mathbb{Z}_{\geq 0}\).
Graduate (set theory)
The natural numbers are the smallest set that contains $0 = \emptyset$ and contains $n \cup \lbrace n \rbrace$ whenever it contains $n$ (the von Neumann construction). Each natural number is the set of all smaller natural numbers.
Notes
This convention is common in set theory, logic, and computer science, and is the one used by the ISO 80000-2 standard. Compare natural numbers starting at 1.
Distinctions involving this term
- Whole numbers vs. natural numbers: “Whole numbers” and “natural numbers” are used synonymously by some others to indicate all nonnegative integers, $0, 1, 2, 3, \dots$. “Whole numbers” is the usual name in primary schools, and “natural numbers” (or “nonnegative integers”) is the usual name in mathematics. “Natural numbers” is, however, used by some authors to indicate the non-negative integers, $1, 2, 3, \dots$ and “whole numbers” is used to indicate all integers, “$\dots, -2, -1, 0, 1, 2, \dots$.