natural numbers

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the natural numbers are those
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s used for counting (as in "there are ''six'' coins on the table") and
ordering Order or ORDER or Orders may refer to: * Orderliness, a desire for organization * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements hav ...
(as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words colloquially used for counting are "
cardinal numbers 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
", and words used for ordering are "
ordinal numbers In set theory illustrating the intersection of two sets 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 s ...
". The natural numbers can, at times, appear as a convenient set of codes (labels or "names"), that is, as what linguists call
nominal number Nominal numbers are categorical, which means that these are numerals used as labels to identify items uniquely. Importantly, the actual values of the number A number is a mathematical object used to counting, count, measurement, measure, and ...
s, forgoing many or all of the properties of being a number in a mathematical sense. Some definitions, including the standard
ISO 80000-2 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization The International Organization for ...
, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers are a basis from which many other number sets may be built by extension: the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s, by including (if not yet in) the
neutral element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
0 and an
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
() for each nonzero natural number ; the
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, by including a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

($\tfrac 1n$) for each nonzero integer (and also the product of these inverses by integers); the
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s by including with the rationals the
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
of (converging)
Cauchy sequences In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose Element (mathematics), elements become arbitrarily close to each other as the sequence progresses.Lang, Serge (1993), Algebra (Third ed.), Reading, Mass ...
of rationals; the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. This chain of extensions make the natural numbers canonically embedded (identified) in the other number systems. Properties of the natural numbers, such as
divisibility In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, are studied in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. Problems concerning counting and ordering, such as partitioning and
enumerations An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the element (mathematics), elements of a Set (mathematics), set. The prec ...
, are studied in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other are ...
. In common language, particularly in primary school education, natural numbers may be called counting numbers to intuitively exclude the negative integers and zero, and also to contrast the discreteness of
counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) ...
to the continuity of
measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of meas ...

—a hallmark characteristic of
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s.

# History

## Ancient roots

The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set. The first major advance in abstraction was the use of
numerals A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''first'' in English) * Numerical di ...
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient
Egyptians Egyptians ( arz, المصريين, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group of people originating from the country of Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinen ...
developed a powerful system of numerals with distinct
hieroglyphs A hieroglyph (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...
for 1, 10, and all powers of 10 up to over 1 million. A stone carving from
Karnak The Karnak Temple Complex, commonly known as Karnak (, which was originally derived from ar, خورنق ''Khurnaq'' "fortified village"), comprises a vast mix of decayed temples A temple (from the Latin Latin (, or , ) is a classical lan ...

, dating back from around 1500 BCE and now at the
Louvre The Louvre ( ), or the Louvre Museum ( ), is the world's largest art museum and a historic monument in Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of Franc ...

in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The
Babylonia Babylonia () was an and based in central-southern which was part of Ancient Persia (present-day and ). A small -ruled state emerged in 1894 BCE, which contained the minor administrative town of . It was merely a small provincial town dur ...
place-value Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral system ...
system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context. A much later advance was the development of the idea that  can be considered as a number, with its own numeral. The use of a 0
digit Digit may refer to: Mathematics and science * Numerical digit, as used in mathematics or computer science ** Arabic numerals, the most common modern representation of numerical digits * Digit (anatomy), one of several most distal parts of a limb ...
in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number. The
Olmec The Olmecs () were the earliest known major Mesoamerica Mesoamerica is a historical region and cultural area in North America North America is a continent entirely within the Northern Hemisphere and almost all within the Western ...
and
Maya civilization The Maya civilization () was a Mesoamerica Mesoamerica is a historical and important region In geography, regions are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human g ...
s used 0 as a separate number as early as the , but this usage did not spread beyond
Mesoamerica Mesoamerica is a historical and important region In geography, regions are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity and th ...
. The use of a numeral 0 in modern times originated with the Indian mathematician
Brahmagupta Brahmagupta ( – ) was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...

in 628 CE. However, 0 had been used as a number in the medieval
computus As a moveable feast, the date of Easter is determined in each year through a calculation known as ''computus'' (Latin for 'computation'). Easter is traditionally celebrated on the first Sunday after the Paschal full moon, being the first full moo ...
(the calculation of the date of Easter), beginning with
Dionysius Exiguus Dionysius Exiguus (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the R ...

in 525 CE, without being denoted by a numeral (standard
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
do not have a symbol for 0). Instead, ''nulla'' (or the genitive form ''nullae'') from ''nullus'', the Latin word for "none", was employed to denote a 0 value. The first systematic study of numbers as
abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental buildin ...

s is usually credited to the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
philosophers
Pythagoras Pythagoras of Samos, or simply ; in () was an ancient and the eponymous founder of . His political and religious teachings were well known in and influenced the philosophies of , , and, through them, . Knowledge of his life is clouded b ...

and
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of ge ...

, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). Independent studies on numbers also occurred at around the same time in
India India, officially the Republic of India (: ), is a country in . It is the by area, the country, and the most populous in the world. Bounded by the on the south, the on the southwest, and the on the southeast, it shares land borders wit ...

, China, and
Mesoamerica Mesoamerica is a historical and important region In geography, regions are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity and th ...
.

## Modern definitions

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
was one of its advocates, as was
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

, who summarized his belief as "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve upon the logical rigor in the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathema ...
. In the 1860s,
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...
suggested a
recursive definition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions were constructed; later still, they were shown to be equivalent in most practical applications. Set-theoretical definitions of natural numbers were initiated by Gottlob Frege, Frege. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem. With all these definitions, it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theory, set theorists and logicians. Other mathematicians also include 0, and computer languages often Zero-based numbering, start from zero when enumerating items like For loop, loop counters and String (computer science), string- or Array data structure, array-elements. On the other hand, many mathematicians have kept the older tradition to take 1 to be the first natural number.

## Notation

Mathematicians use or $\mathbb N$ to refer to the Set (mathematics), set of all natural numbers. The existence of such a set is established in set theory. Older texts have also occasionally employed as the symbol for this set. Since different properties are customarily associated to the tokens and (e.g., neutral elements for addition and multiplications, respectively), it is important to know which version of ''natural numbers'' is employed in the case under consideration. This can be done by explanation in prose, by explicitly writing down the set, or by qualifying the generic identifier with a super- or subscript, for example, like this: * Naturals without zero: $\=\mathbb^*= \mathbb N^+=\mathbb_0\smallsetminus\ = \mathbb_1$ * Naturals with zero: $\;\=\mathbb_0=\mathbb N^0=\mathbb^*\cup\$ Alternatively, since the natural numbers naturally form a subset of the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s (often they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a subscript (or superscript) "0" is added in the former case, and a superscript "" is added in the latter case: :$\ = \=\mathbb Z^+= \mathbb_$ :$\ = \=\mathbb Z^_=\mathbb_$

# Properties

Given the set $\mathbb$ of natural numbers and the successor function $S \colon \mathbb \to \mathbb$ sending each natural number to the next one, one can define Addition in N, addition of natural numbers recursively by setting and for all , . Then is a commutative monoid with identity element 0. It is a free object, free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group (mathematics), group. The smallest group containing the natural numbers is the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s. If 1 is defined as , then . That is, is simply the successor of .

## Multiplication

Analogously, given that addition has been defined, a multiplication operator $\times$ can be defined via and . This turns into a free commutative monoid with identity element 1; a generator set for this monoid is the set of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s.

## Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in the distributivity, distribution law: . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that is not closure (mathematics), closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that is ''not'' a ring (mathematics), ring; instead it is a semiring (also known as a ''rig''). If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with and .

## Order

In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting if and only if there exists another natural number where . This order is compatible with the arithmetical operations in the following sense: if , and are natural numbers and , then and . An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as (omega).

## Division

In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of ''division with remainder'' or Euclidean division is available as a substitute: for any two natural numbers and with there are natural numbers and such that :$a = bq + r \text r < b.$ The number is called the ''quotient'' and is called the ''remainder'' of the division of by . The numbers and are uniquely determined by and . This Euclidean division is key to the several other properties (
divisibility In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
), algorithms (such as the Euclidean algorithm), and ideas in number theory.

## Algebraic properties satisfied by the natural numbers

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: * Closure (mathematics), Closure under addition and multiplication: for all natural numbers and , both and are natural numbers. * Associativity: for all natural numbers , , and , and . * Commutativity: for all natural numbers and , and . * Existence of identity elements: for every natural number ''a'', and . * Distributivity of multiplication over addition for all natural numbers , , and , . * No nonzero zero divisors: if and are natural numbers such that , then or (or both).

## Infinity

The set of natural numbers is an infinite set. By definition, this kind of infinity is called countably infinite, countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-nought ().

# Generalizations

Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers. * A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have equinumerosity, the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be ''countable set, countably infinite'' and to have cardinality Aleph number#Aleph-null, aleph-null (). * Natural numbers are also used as Ordinal numbers (linguistics), linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ; this is also the ordinal number of the set of natural numbers itself. The least ordinal of cardinality (that is, the Von Neumann cardinal assignment, initial ordinal of ) is but many well-ordered sets with cardinal number have an ordinal number greater than . For finite set, finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence. A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction. Georges Reeb used to claim provocatively that ''The naïve integers don't fill up'' . Other generalizations are discussed in the article on numbers.

# Formal definitions

## Peano axioms

Many properties of the natural numbers can be derived from the five Peano axioms: # 0 is a natural number. # Every natural number has a successor which is also a natural number. # 0 is not the successor of any natural number. # If the successor of $x$ equals the successor of $y$, then $x$ equals $y$. # The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number. These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of $x$ is $x + 1$. Replacing axiom 5 by an axiom schema, one obtains a (weaker) first-order theory called ''Peano arithmetic''.

## Constructions based on set theory

### Von Neumann ordinals

In the area of mathematics called set theory, a specific construction due to John von Neumann attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s. defines the natural numbers as follows: * Set , the empty set, * Define for every set . is the successor of , and is called the successor function. * By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be ''inductive''. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms. * It follows that each natural number is equal to the set of all natural numbers less than it: :*, :*, :*, :*, :*, etc. With this definition, a natural number is a particular set with elements, and if and only if is a subset of . The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the well-ordered set of all smaller ordinals." Also, with this definition, different possible interpretations of notations like (-tuples versus mappings of into ) coincide. Even if one finitism, does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

### Zermelo ordinals

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows: * Set * Define , * It then follows that :*, :*, :*, :*, etc. :Each natural number is then equal to the set containing just the natural number preceding it. This is the definition of Zermelo ordinals. Unlike von Neumann's construction, the Zermelo ordinals do not account for infinite ordinals.

* Benacerraf's identification problem * Canonical representation of a positive integer * Countable set * Number#Classification for other number systems (rational, real, complex etc.) * Ordinal number * Set-theoretic definition of natural numbers

# Bibliography

* * * * ** ** * * * * * * * * * * * * * * – English translation of .