set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...

, several ways have been proposed to construct the

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

. These include the representation via von Neumann ordinals, commonly employed in

axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...

, 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 philos ...

and by

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

Definition as von Neumann ordinals

In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting be the

empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

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 A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

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 nea ...

. 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 ...

in ZF set theory. The set ''N'' and its elements, when constructed this way, are an initial part of the von Neumann ordinals. Quine refer to these sets as "counter sets".W. V. O. Quine, ''Mathematical Logic'' (1981), p.247. Harvard University Press, 0-674-55451-5.

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 ...

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. A simpler example is equ ...

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, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...

—this is sometimes known as

Hume's principle Hume's principle or HP says that, given two collections of objects \mathcal F and \mathcal G with properties F and G respectively, the number of objects with property F is equal to the number of objects with property G if and only if there is a ...

. This definition works in

type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...

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

New Foundations In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Definition The well-formed fo ...

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. However, cardinals can be defined in ZF using

Scott's trick In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity bu ...

. 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. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, 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 ...

. The Russell paradox proved this system inconsistent, but George Boolos (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 Bachelor of Arts, BA from Rice University in 1 ...

(2004) show how to repair it.

References

* Anderson, D. J., and

, 2004, "Frege, Boolos, and Logical Objects," ''Journal of Philosophical Logic 33'': 1–26. * George Boolos, 1998. ''Logic, Logic, and Logic''. * * * 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. *

Citations

{{reflist

External links

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...

: *
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