section_path: Formal definitions/Set-theoretic definition
section_name: Set-theoretic definition
paragraph_text: Set-theoretic definition
  
  Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence ". This does not work in all set theories, as such an equivalence class would not be a set (because of Russell's paradox). The standard solution is to define a particular set with n elements that will be called the natural number n.
  
  The following definition was first published by John von Neumann, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number, the sets considered below are sometimes called von Neumann ordinals.
  
  The definition proceeds as follows:
  
  It follows that the natural numbers are defined iteratively as follows:
  
  It can be checked that the natural numbers satisfy the Peano axioms.
  
  With this definition, given a natural number n, the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S. This formalizes the operation of counting the elements of S. Also, n ≤ m if and only if n is a subset of m. In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order.
  
  It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all ordinal numbers, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
  
  If one does not accept the axiom of infinity, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
  
  There are other set theoretical constructions. In particular, Ernst Zermelo provided a construction that is nowadays only of historical interest, and is sometimes referred to as Zermelo ordinals. It consists in defining 0 as the empty set, and S (a) = { a }.
  
  With this definition each nonzero natural number is a singleton set. So, the property of the natural numbers to represent cardinalities is not directly accessible; only the ordinal property (being the n th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.