set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, several ways have been proposed to construct the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s. These include the representation via

von Neumann ordinal In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...

s, commonly employed in

axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, and a system based on equinumerosity that was proposed by

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...

and by

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...

Definition as von Neumann ordinals

In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined

recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

by letting be the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

and for each ''n''. In this way for each natural number ''n''. This definition has the property that ''n'' is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

with ''n'' elements. The first few numbers defined this way are: :

\begin
   0 &  = \      &&  = \varnothing,\\
   1 &  = \     &&  = \,\\
   2 &  = \   &&  = \,\\
   3 &  = \ &&  = \.
 \end

The set ''N'' of natural numbers is defined in this system as the smallest set containing 0 and closed under the

successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...

''S'' defined by . The structure is a model of the

Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

. The existence of the set ''N'' is equivalent to the

axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...

in ZF set theory. The set ''N'' and its elements, when constructed this way, are an initial part of the von Neumann ordinals.

Frege and Russell

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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

of finite sets under the

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

—this is sometimes known as

Hume's principle Hume's principle or HP says that the number of ''F''s is equal to the number of ''G''s if and only if there is a one-to-one correspondence (a bijection) between the ''F''s and the ''G''s. HP can be stated formally in systems of second-order logic. ...

. This definition works in

type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

, and in set theories that grew out of type theory, such as

New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of '' Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundatio ...

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 class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...

es rather than sets. 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() = Card() Definition: the successor of a cardinal K is the cardinal K + 1 Theorem: the natural numbers satisfy Peano’s axioms

Hatcher

William S. Hatcher (1982) derives Peano's axioms from several foundational systems, including ZFC and

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

, and from the system of Frege's ''Grundgesetze der Arithmetik'' using modern notation and

natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ax ...

. The

Russell paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...

proved this system inconsistent, but

George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek- Jewish descent. He graduated with an A.B. ...

(1998) and David J. Anderson and

Edward Zalta Edward Nouri Zalta (; born March 16, 1952) is an American philosopher who is a senior research scholar at the Center for the Study of Language and Information at Stanford University. He received his BA at Rice University in 1975 and his PhD fro ...

(2004) show how to repair it.

References

* Anderson, D. J., and

, 2004, "Frege, Boolos, and Logical Objects," ''Journal of Philosophical Logic 33'': 1–26. *

, 1998. ''Logic, Logic, and Logic''. * {{Cite book , first = Derek , last = Goldrei , title = Classic Set Theory , publisher =

Chapman & Hall Chapman & Hall is an Imprint (trade name), imprint owned by CRC Press, originally founded as a United Kingdom, British publishing house in London in the first half of the 19th century by Edward Chapman (publisher), Edward Chapman and William Hall ...

, year = 1996 *

Abraham Fraenkel Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. ...

, 1968 (1953). ''Abstrast Set Theory''. North Holland, Amsterdam, 4th edtition. * Hatcher, William S., 1982. ''The Logical Foundations of Mathematics''. Pergamon. In this text, S refers to the Peano axioms. * Holmes, Randall, 1998.
Elementary Set Theory with a Universal Set
'. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this introduction to NFU via the web. Copyright is reserved. *

Patrick Suppes Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology ...

, 1972 (1960). ''Axiomatic Set Theory''. Dover.

External links

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...

: *
Quine's New Foundations
— by Thomas Forster. *
Alternative axiomatic set theories
— by Randall Holmes. * McGuire, Gary,
What are the Natural Numbers?
* Randall Holmes

Basic concepts in infinite set theory

Definition as von Neumann ordinals

Frege and Russell

Hatcher

See also

References

External links