Inset theory, several ways have been proposed to construct thenatural numbers. These include the representation viavon Neumann ordinals, commonly employed inaxiomatic set theory, and a system based onequinumerosity that was proposed byGottlob Frege and byBertrand Russell.
InZermelo–Fraenkel (ZF) set theory, the natural numbers are definedrecursively by letting0 = {} be theempty set andn + 1 (the successor function) =n ∪ {n} for eachn. In this wayn = {0, 1, …,n − 1} for each natural numbern. This definition has the property thatn is aset withn elements. The first few numbers defined this way are: (Goldrei 1996)
The setN of natural numbers is defined in this system as the smallest set containing 0 and closed under thesuccessor functionS defined byS(n) =n ∪ {n}. Thestructure⟨N, 0,S⟩ is a model of thePeano axioms (Goldrei 1996). The existence of the setN is equivalent to theaxiom of infinity in ZF set theory.
The setN and its elements, when constructed this way, are an initial part of the von Neumann ordinals.W. V. O. Quine refers to these sets as "counter sets".[1]
Gottlob Frege and Bertrand Russell each proposed defining a natural numbern as the collection of all sets withn elements. More formally, a natural number is anequivalence class of finite sets under theequivalence relation ofequinumerosity. This definition may appear circular, but it is not, because equinumerosity can be defined in alternate ways, for instance by saying that two sets are equinumerous if they can be put intoone-to-one correspondence—this is sometimes known asHume's principle.
This definition works intype theory, and in set theories that grew out of type theory, such asNew Foundations and related systems. However, it does not work in the axiomatic set theoryZFC nor in certain related systems, because in such systems the equivalence classes under equinumerosity areproper classes rather than sets. However, cardinals can be defined in ZF usingScott's trick.
For enabling natural numbers to form a set, equinumerous classes are replaced by special sets, named cardinal. The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice).[2]
Axiom of cardinality: The sets A and B are equinumerous if and only if Card(A) = Card(B)
Definition: the sum of cardinals K and L such as K= Card(A) and L = Card(B) where the sets A and B are disjoint, is Card (A ∪ B).
The definition of a finite set is given independently of natural numbers:[3]
Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order.
Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n.
0 = Card (∅)
1 = Card({A}) = Card({∅})
Definition: the successor of a cardinal K is the cardinal K + 1
Theorem: the natural numbers satisfy Peano’s axioms
William S. Hatcher (1982) derives Peano's axioms from several foundational systems, includingZFC andcategory theory, and from the system of Frege'sGrundgesetze der Arithmetik using modern notation andnatural deduction. TheRussell paradox proved this system inconsistent, butGeorge Boolos (1998) and David J. Anderson andEdward Zalta (2004) show how to repair it.
