Set-theoretic definition of natural numbers

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The oldest definition

The original set theoretical definition of the natural numbers is generally ascribed to Frege and Russell. An informal way to put this definition is that each concrete natural number n is defined as the set of all sets with n elements. This appears circular but is not; it is necessary to be more explicit to see this.

Define 0 as <math>\{\{\}\}</math> (obviously the set of all sets with 0 elements). For any set A, define <math>\sigma(A)</math> as <math>\{x \cup \{y\} \mid x \in A \wedge y \not\in x\}</math>, the set of all sets obtainable by adding one new element to a set in A. We view <math>\sigma</math> as a generalization of the successor function, and define 1 as <math>\sigma(0)</math>, 2 as <math>\sigma(1)</math>, 3 as <math>\sigma(2)</math>, and so forth. This definition has the desired effect: the 3 we have just defined actually is the set of all sets with three elements.

If the universe V has a finite cardinality n, we would have <math>n+1 = \sigma(V)= \emptyset</math> and of course <math>\sigma(\emptyset)=\emptyset</math>; the sequence of natural numbers would come to an end. An Axiom of Infinity averts this, and given an axiom of infinity the Frege-Russell natural numbers satisfy the Peano axioms. The set of natural numbers can be defined as the intersection of all sets containing 0 and closed under <math>\sigma</math>.

Unfortunately, this beautiful definition does not work in ZFC or related systems of axiomatic set theory because the collections used to represent the natural numbers are too large to be sets. This definition does work in type theory and in New Foundations and related systems.

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The usual definition

A set-theoretic definition of the natural numbers which does work in ZFC and related theories was proposed by John von Neumann. It allows the discussion of natural numbers in a system based (as modern mathematics is) on axiomatic set theory.

Neumann proposed the following definition:

Define the empty set to be zero.
Define the successor of n as n ∪ {n}

The set N of all natural numbers is then guaranteed to exist by axiom of infinity. It can easily be shown that the above definition satisfies the Peano axioms. It also (in contrast to some alternative definitions) has the property that each natural number n is a set with exactly n elements: {0,1,2,...,n-1}

See Peano arithmetic.

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