The natural numbers \(0,~1,~2,~3,\ldots\), otherwise known as the non-negative integers^{1}, are the numbers familiar even to young children. It is the natural numbers that we use for the very basic mathematical purpose of counting things (see cardinal numbers). We also can use them to order things (see ordinal numbers).
From the formal point of view, natural numbers can be constructed using sets. In particular, we can use the concept of the empty set \(\emptyset\) and call it \(0\), then construct a set containing this empty set and call it \(1\), then construct a set containing the empty set and also the previous set and call it \(2\), etc.:
\[\begin{array}{cclcl}0&:=&\emptyset&=&\emptyset\\1&:=&\left\{0\right\}&=&\left\{\emptyset\right\}\\2&:=&\left\{0,~1\right\}&=&\left\{\emptyset,~\left\{\emptyset\right\}\right\}\\3&:=&\left\{0,~1,~2\right\}&=&\left\{\emptyset,~\left\{\emptyset,~\left\{\emptyset\right\}\right\}\right\}\\\vdots&&\vdots&&\vdots\\\end{array}\]
Another way to define natural numbers is to use a suitable axiomatic system. One of such axiomatic systems are the Peano Axioms.
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Please note: Another common way is to understand the set of natural numbers is as positive integers, excluding the number 0. In BookOfProofs, natural numbers are always including 0, by convention. ↩