Gottlob Frege and Bertrand Russell each proposed defining a natural number
n as the collection of all sets with
n elements. More formally, a natural number is an
equivalence class of finite sets under the
equivalence relation of
equinumerosity. 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 into
one-to-one correspondence—this is sometimes known as
Hume's principle. This definition works in
type theory, and in set theories that grew out of type theory, such as
New Foundations and related systems. However, it does not work in the axiomatic set theory
ZFC nor in certain related systems, because in such systems the equivalence classes under equinumerosity are
proper classes rather than sets. However, cardinals can be defined in ZF using
Scott'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). 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: 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 == Hatcher ==