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Given the Zermelo-Fraenkel axioms of set theory, one can prove that there exists an inductive set  such that
 . The natural numbers
 are then defined to be the intersection of all subsets of  which are inductive sets and contain the empty set as an element.
The first few natural numbers are:
Note that the set 0 has zero elements, the set  has one element, the set  has two elements, etc. Informally, the set  is the set consisting of the  elements
 , and  is both a subset of
 and an element of
 .
In some contexts (most notably, in number theory), it is more convenient to exclude 0 from the set of natural numbers, so that
 . When it is not explicitly specified, one must determine from context whether 0 is being considered a natural number or not.
Addition of natural numbers is defined inductively as follows:
Multiplication of natural numbers is defined inductively as follows:
The natural numbers form a monoid under either addition or multiplication. There is an ordering relation on the natural numbers, defined by:  if
 .
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