Natural number: Difference between revisions

Latest revision as of 11:58, 3 November 2025

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Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0.Template:Efn^[1] The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used.^[2]^[3] The set of the natural numbers is commonly denoted by a bold Template:Math or a blackboard bold Template:Tmath.

The natural numbers are used for counting, and for labeling the result of a count, like "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, like "the third day of the month", in which case they are called ordinal numbers. Natural numbers may also be used to label, like the jersey numbers of a sports team; in this case, they have no specific mathematical properties and are called nominal numbers.^[4]

Two natural operations are defined on natural numbers, addition and multiplication. Arithmetic is the study of the ways to perform these operations. Number theory is the study of the properties of these operations and their generalizations. Much of combinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.

Many number systems are built from the natural numbers and contain them. For example, the integers are made by including 0 and negative numbers. The rational numbers add fractions, and the real numbers add all infinite decimals. Complex numbers add the [[Imaginary unit|square root of Template:Math]].^[5] This makes up natural numbers as foundational for all mathematics.^[6]

Terminology and notation

The term natural numbers has two common definitions: either Template:Math or Template:Math. Because there is no universal convention, the definition can be chosen to suit the context of use.^[1]^[7] To eliminate ambiguity, the sequences Template:Math and Template:Math are often called the positive integers and the non-negative integers, respectively.

The phrase whole numbers is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative.^[8]^[2] In primary education, counting numbers usually refer to the natural numbers starting at 1,^[3] though this definition can vary.^[9]^[10]

The set of all natural numbers is typically denoted Template:Math or in blackboard bold as $ℕ .$ ^[7]^[11]Template:Efn Whether 0 is included is often determined by the context but may also be specified by using $ℕ$ or $ℤ$ (the set of all integers) with a subscript or superscript. Examples include $ℕ_{1}$ ,^[12] or $ℤ^{+}$ ^[13] (for the set starting at 1) and $ℕ_{0}$ ^[14] or $ℤ^{0 +}$ ^[15] (for the set including 0).

Intuitive concept

An intuitive and implicit understanding of natural numbers is developed naturally through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the size of a collection; and a position in a sequence.

Size of a collection

Natural numbers can be used to answer questions like: "how many apples are on the table?".^[16] A natural number used in this way describes a characteristic of a collection of objects. This characteristic, the size of a collection is called cardinality and a natural number used to describe or measure it is called a cardinal number.

File:Cardinality apples and oranges.svg

A group of apples and group of oranges with the same cardinality.

Two collections have the same size or cardinality if there is a one-to-one correspondence between the objects in each collection to the objects in the other. For example, in the image to the right every apple can be paired off with one orange and every orange can be paired off with one apple. From this, even without counting or using numbers it can be seen that the group of apples has the same cardinality as the group of oranges, meaning they are both assigned the same cardinal number.

The natural number 3 is the thing used for the particular cardinal number described above and for the cardinal number of any other collection of objects that could be paired off in the same way to one of these groups.

Position in a sequence

The natural numbers have a fixed progression, which is the familiar sequence beginning with 1, 2, 3, and so on. A natural number can be used to denote a specific position in any other sequence, in which case it is called an ordinal number. To have a specific position in a sequence means to come either before or after every other position in the sequence in a defined way, which is the concept of order.

The natural number 3 then is the thing that comes after 2 and 1, and before 4, 5 and so on. The number 2 is the thing that comes after 1, and 1 is the first element in the sequence. Each number represents the relation that position bears to the rest of the infinite sequence.^[17]

Counting

The process of counting involves both the cardinal and ordinal use of the natural numbers and illustrates the way the two fit together. To count the number of objects in a collection, each object is paired off with a natural number, usually by mentally or verbally saying the name of the number and assigning it to a particular object. The numbers must be assigned in order starting with 1 (they are ordinal) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. When all of the objects have been assigned a number, the ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the whole collection.

History

Ancient roots

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File:Ishango bone (cropped).jpg

The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)^[18]^[19]^[20] is believed to have been used 20,000 years ago for natural number arithmetic.

The most primitive method of representing a natural number is to use one's fingers, as in finger counting. Putting down a tally mark for each object is another primitive method. 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 to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak, dating back from around 1500 BCE and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value 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.^[21]

A much later advance was the development of the idea that Template:Num can be considered as a number, with its own numeral. The use of a 0 digit 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.Template:Efn The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE, but this usage did not spread beyond Mesoamerica.^[22]^[23] The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae) from Script error: No such module "Lang"., the Latin word for "none", was employed to denote a 0 value.^[24]

The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.Template:Efn Euclid, 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).^[25] However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other.^[26]

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.^[27]

Emergence as a term

Nicolas Chuquet used the term progression naturelle (natural progression) in 1484.^[28] The earliest known use of "natural number" as a complete English phrase is in 1763.^[29]^[30] The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.^[30]

Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.^[31] In 1889, Giuseppe Peano used N for the positive integers and started at 1,^[32] but he later changed to using N₀ and N₁.^[33] Historically, most definitions have excluded 0,^[30]^[34]^[35] but many mathematicians such as George A. Wentworth, Bertrand Russell, Nicolas Bourbaki, Paul Halmos, Stephen Cole Kleene, and John Horton Conway have preferred to include 0.^[36]^[30] This approach gained wider adoption in the 1960s^[30] and was formalized in ISO 31-11 (1978), which defines natural numbers to include zero, a convention retained in the current ISO 80000-2 standard.^[37]

Formal construction

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.^[38] Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".Template:Efn

The constructivists saw a need to improve upon the logical rigor in the foundations of mathematics.Template:Efn In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.

Set-theoretical definitions of natural numbers were initiated by 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.^[39]

In 1881, Charles Sanders Peirce provided the first axiomatization of natural-number arithmetic.^[40]^[41] In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic,^[42] and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method (Template:Langx). 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.^[43] Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.^[44]

Formal definitions

Formal definitions of the natural numbers take the existing, intuitive notion of natural numbers and the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. The two standard methods for doing this are: the Peano axioms; and set theory.

The Peano axioms (named for Giuseppe Peano) do not explicitly define what the natural numbers are, but instead comprise a list of statements or axioms that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particular set, in which a set can be generally understood as a collection of distinct objects or elements. While the two approaches are different, they are consistent in that the natural number sets collectively satisfy the Peano axioms.

Peano axioms

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The five Peano axioms are the following:^[45]Template:Efn

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 $x + 1$ .

Set-theoretic definition

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In set theory each natural number Template:Mvar is defined as an explicitly defined set, whose elements allow counting the elements of other sets. A variety of different constructions have been proposed, however the standard solution (due to John von Neumann)^[46] is to define each natural number Template:Mvar as a set containing Template:Mvar elements in the following way:

Call Template:Math, the empty set.
Define the successor Template:Math of any set Template:Mvar by Template:Math.
By the axiom of infinity, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be inductive. The intersection of all inductive sets is still an inductive set.
This intersection is the set of the natural numbers.

This produces an iterative definition of the natural numbers satisfying the Peano axioms, sometimes called von Neumann ordinals:

In this definition each natural number is equal to the set of all natural numbers less than it. Given a natural number Template:Math, the sentence "a set Template:Mvar has Template:Mvar elements" can be formally defined as "there exists a bijection from Template:Mvar to Template:Mvar." This formalizes the operation of counting the elements of Template:Mvar. Also, Template:Math if and only if Template:Math is a subset of Template:Math. In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order.

Another construction sometimes called Template:Vanchor^[47] defines Template:Math and Template:Math and is now largely only of historical interest.

Properties

This section uses the convention $ℕ = ℕ_{0} = ℕ^{*} \cup {0}$ .

Addition

Given the set $ℕ$ of natural numbers and the successor function $S : ℕ \to ℕ$ sending each natural number to the next one, one can define addition of natural numbers recursively by setting Template:Math and Template:Math for all Template:Math, Template:Math. Thus, Template:Math, Template:Math, and so on. The algebraic structure $(ℕ, +)$ is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers.

If 1 is defined as Template:Math, then Template:Math. That is, Template:Math is simply the successor of Template:Math.

Multiplication

Analogously, given that addition has been defined, a multiplication operator $\times$ can be defined via Template:Math and Template:Math. This turns $(ℕ^{*}, \times)$ into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.

Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in the distribution law: Template:Math. 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 $ℕ$ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that $ℕ$ is not a 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 Template:Math and Template:Math. Furthermore, $(ℕ^{*}, +)$ has no identity element.

Order

In this section, juxtaposed variables such as Template:Math indicate the product Template:Math,^[48] and the standard order of operations is assumed.

A total order on the natural numbers is defined by letting Template:Math if and only if there exists another natural number Template:Math where Template:Math. This order is compatible with the arithmetical operations in the following sense: if Template:Math, Template:Math and Template:Math are natural numbers and Template:Math, then Template:Math and Template:Math.

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 Template:Math (omega).

Division

In this section, juxtaposed variables such as Template:Math indicate the product Template:Math, 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 Template:Math and Template:Math with Template:Math there are natural numbers Template:Math and Template:Math such that

a = b q + r and r < b .

The number Template:Math is called the quotient and Template:Math is called the remainder of the division of Template:Math by Template:Math. The numbers Template:Math and Template:Math are uniquely determined by Template:Math and Template:Math. This Euclidean division is key to the several other properties (divisibility), 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 under addition and multiplication: for all natural numbers Template:Math and Template:Math, both Template:Math and Template:Math are natural numbers.^[49]
Associativity: for all natural numbers Template:Math, Template:Math, and Template:Math, Template:Math and Template:Math.^[50]
Commutativity: for all natural numbers Template:Math and Template:Math, Template:Math and Template:Math.^[51]
Existence of identity elements: for every natural number Template:Math, Template:Math and Template:Math.
- If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number Template:Math, Template:Math. However, the "existence of additive identity element" property is not satisfied
Distributivity of multiplication over addition for all natural numbers Template:Math, Template:Math, and Template:Math, Template:Math.
No nonzero zero divisors: if Template:Math and Template:Math are natural numbers such that Template:Math, then Template:Math or Template:Math (or both).

Generalizations

Natural numbers are broadly used in two ways: to quantify and to order. A number used to represent the quantity of objects in a collection ("there are 6 coins on the table") is called a cardinal numeral, while a number used to order individual objects within a collection ("she finished 6th in the race") is an ordinal numeral.

These two uses of natural numbers apply only to finite sets. Georg Cantor discovered at the end of the 19th century that both uses of natural numbers can be generalized to infinite sets, but that they lead to two different concepts of "infinite" numbers, the cardinal numbers and the ordinal numbers.

Other generalizations of natural numbers are discussed in Template:Section link.

Notes

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References

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Bibliography

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External links

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Natural number: Difference between revisions

Latest revision as of 11:58, 3 November 2025

Contents

Terminology and notation

Intuitive concept

Size of a collection

Position in a sequence

Counting

History

Ancient roots

Emergence as a term

Formal construction

Formal definitions

Peano axioms

Set-theoretic definition

Properties

Addition

Multiplication

Relationship between addition and multiplication

Order

Division

Algebraic properties satisfied by the natural numbers

Generalizations

See also

Notes

References

Bibliography

External links

Navigation menu

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-{{Short description|Number used for counting}}
+{{Short description|Number used for counting}}{{pp-vandalism|small=yes}}
-{{pp-pc}}
 {{Use dmy dates|date=May 2021}}
 [[File:Three Baskets with Apples.svg|right|thumb|upright|Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...]]
-In [[mathematics]], the '''natural numbers''' are the numbers [[0]], [[1]], [[2]], [[3]], and so on, possibly excluding 0.<ref name="Enderton"/> Some start counting with 0, defining the natural numbers as the '''non-negative integers''' {{math|1=0, 1, 2, 3, ...}}, while others start with 1, defining them as the '''positive integers''' {{nobr|{{math|1, 2, 3, ...}} .{{efn|See {{section link|#Emergence as a term}}}} }} Some authors acknowledge both definitions whenever convenient.<ref name=":1">{{cite web |last=Weisstein |first=Eric W. |title=Natural Number |url=https://mathworld.wolfram.com/NaturalNumber.html |access-date=11 August 2020 |website=mathworld.wolfram.com |language=en}}</ref> Sometimes, the '''whole numbers''' are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the [[integer]]s, including negative integers.<ref>{{cite dictionary |first1=Jack G. |last1=Ganssle |first2=Michael |last2=Barr |name-list-style=amp |year=2003 |dictionary=Embedded Systems Dictionary |isbn=978-1-57820-120-4 |title=integer |pages=138 (integer), 247 (signed integer), & 276 (unsigned integer) |publisher=Taylor & Francis |via=Google Books |url=https://books.google.com/books?id=zePGx82d_fwC |access-date=28 March 2017 |url-status=live |archive-url=https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC |archive-date=29 March 2017}}</ref> The '''counting numbers''' are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1.<ref name=MathWorld_CountingNumber>{{MathWorld|title=Counting Number|id=CountingNumber}}</ref>
+In [[mathematics]], the '''natural numbers''' are the numbers [[0]], [[1]], [[2]], [[3]], and so on, possibly excluding 0.{{efn|It depends on authors and context whether 0 is considered a natural number.}}<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> The terms '''positive integers''', '''non-negative integers''', '''whole numbers''', and '''counting numbers''' are also used.<ref name="Cooke2000">{{cite book |last1=Cooke |first1=Heather |url=https://www.google.com/books/edition/Primary_Mathematics/yNCL5iTJqLQC?hl=en&gbpv=1&dq=%22counting%20numbers%22%20%22positive%20integers%22&pg=PA14&printsec=frontcover |title=Primary Mathematics |date=26 October 2000 |publisher=SAGE |isbn=978-1-84787-949-3 |page=14 |language=en}}</ref><ref name="Zegarelli2014">{{cite book |last1=Zegarelli |first1=Mark |url=https://www.google.com/books/edition/Basic_Math_and_Pre_Algebra_For_Dummies/V8nDAgAAQBAJ?hl=en&gbpv=1&dq=%22counting%20numbers%22&pg=PA21&printsec=frontcover |title=Basic Math and Pre-Algebra For Dummies |date=28 January 2014 |publisher=John Wiley & Sons |isbn=978-1-118-79199-8 |page=21 |language=en |quote=Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.}}</ref> The [[set (mathematics)|set]] of the natural numbers is commonly denoted by a bold {{math|'''N'''}} or a [[blackboard bold]] {{tmath|\N}}.
-The natural numbers are used for counting<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> things, like "there are ''six'' coins on the table", in which case they are called ''[[cardinal number]]s''. They are also used to put things in order,<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> like "this is the ''third'' largest city in the country", which are called ''[[ordinal number]]s''. Natural numbers are also used as labels, like [[Number (sports)|jersey numbers]] on a sports team, where they serve as ''[[nominal number]]s'' and do not have mathematical properties.<ref>{{cite journal |last1=Woodin |first1=Greg |first2=Bodo |last2=Winter |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48|number=6 |year=2024 |article-number=e13471 |doi=10.1111/cogs.13471 |doi-access=free|pmid=38895756 |pmc=11475258 }}</ref>
+The natural numbers are used for counting,<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> and for labeling the result of a count, like "there are ''seven'' days in a week", in which case they are called ''[[cardinal numeral|cardinal number]]s''. They are also used to label places in an ordered series,<!-- Please, do not link these words that are used in their common language meanings, and not in any technical meaning --> like "the ''third'' day of the month", in which case they are called ''[[ordinal numeral|ordinal number]]s''. Natural numbers may also be used to label, like the [[Number (sports)|jersey numbers]] of a sports team; in this case, they have no specific mathematical properties and are called ''[[nominal number]]s''.<ref>{{cite journal |last1=Woodin |first1=Greg |first2=Bodo |last2=Winter |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48|number=6 |year=2024 |article-number=e13471 |doi=10.1111/cogs.13471 |doi-access=free|pmid=38895756 |pmc=11475258 }}</ref>
-The natural numbers form a [[set (mathematics)|set]], commonly symbolized as a bold {{math|'''N'''}} or [[blackboard bold]] {{tmath|\N}}. Many other [[number set]]s are built from the natural numbers. For example, the [[integer]]s are made by adding 0 and negative numbers. The [[rational number]]s add fractions, and the [[real number]]s add all infinite decimals. [[Complex number]]s add the [[Imaginary unit|square root of {{math|−1}}]]. This chain of extensions canonically [[Embedding|embeds]] the natural numbers in the other number systems.<ref>{{harvtxt|Mendelson|2008|page=x}} says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."</ref><ref>{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."</ref>
+Two natural [[operation (mathematics)|operation]]s are defined on natural numbers, [[addition]] and [[multiplication]]. [[Arithmetic]] is the study of the ways to perform these operations. [[Number theory]] is the study of the properties of these operations and their generalizations. Much of [[combinatorics]] involves counting mathematical objects, patterns and structures that are defined using natural numbers.
-Natural numbers are studied in different areas of math. [[Number theory]] looks at things like how numbers divide evenly ([[divisibility]]), or how [[prime number]]s are spread out. [[Combinatorics]] studies counting and arranging numbered objects, such as [[Partition (number theory)|partition]]s and [[Enumerative combinatorics|enumerations]].
+Many [[number system]]s are built from the natural numbers and contain them. For example, the [[integer]]s are made by including 0 and negative numbers. The [[rational number]]s add fractions, and the [[real number]]s add all infinite decimals. [[Complex number]]s add the [[Imaginary unit|square root of {{math|−1}}]].<ref>{{harvtxt|Mendelson|2008|page=x}} says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."</ref>  This makes up natural numbers as foundational for all mathematics.<ref>{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."</ref>
+== Terminology and notation ==
+The term ''natural numbers'' has two common definitions: either {{math|1=0, 1, 2, ...}} or {{math|1, 2, 3, ...}}. Because there is no universal convention, the definition can be chosen to suit the context of use.<ref name="Enderton" /><ref name=":1">{{cite web |last=Weisstein |first=Eric W. |title=Natural Number |url=https://mathworld.wolfram.com/NaturalNumber.html |access-date=11 August 2020 |website=mathworld.wolfram.com |language=en}}</ref> To eliminate ambiguity, the sequences {{math|1, 2, 3, ...}} and {{math|1=0, 1, 2, ...}} are often called the '''positive integers''' and the '''non-negative integers''', respectively.
+The phrase '''whole numbers''' is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative.<ref>{{cite dictionary |year=2003 |title=integer |dictionary=Embedded Systems Dictionary |publisher=Taylor & Francis |url=https://books.google.com/books?id=zePGx82d_fwC |access-date=28 March 2017 |pages=138 (integer), 247 (signed integer), & 276 (unsigned integer) |isbn=978-1-57820-120-4 |url-status=live |archive-url=https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC |archive-date=29 March 2017 |last2=Barr |first2=Michael |name-list-style=amp |first1=Jack G. |last1=Ganssle |via=Google Books}}</ref><ref name="Cooke2000" /> In primary education, '''counting numbers''' usually refer to the natural numbers starting at 1,<ref name="Zegarelli2014" /> though this definition can vary.<ref>{{cite book |last1=Rice |first1=Harris |url=https://www.google.com/books/edition/The_Mathematics_Teacher/2kwH8pZKLLEC?hl=en&gbpv=1&dq=%22counting%20numbers%22&pg=PA393&printsec=frontcover |title=The Mathematics Teacher |date=1922 |publisher=National Council of Teachers of Mathematics |page=393 |language=en |chapter=Errors in computations and the rounded number |quote=A counting number is the number given in answer to the question "How many?" In this class of numbers belongs zero and positive integers/}}</ref><ref name="MathWorld_CountingNumber">{{MathWorld|title=Counting Number|id=CountingNumber}}</ref>
+The [[Set (mathematics)|set]] of all natural numbers is typically denoted {{math|'''N'''}} or in [[blackboard bold]] as <math>\mathbb N.</math><ref name=":1" /><ref>{{cite web |title=Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions |url=https://functions.wolfram.com/Notations/1/ |access-date=27 July 2020 |website=functions.wolfram.com}}</ref>{{Efn|Older texts have occasionally employed {{math|''J''}} as the symbol for this set.<ref>{{cite book |url=https://archive.org/details/1979RudinW |title=Principles of Mathematical Analysis |last=Rudin |first=W. |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8 |location=New York |page=25}}</ref>}} Whether 0 is included is often determined by the context but may also be specified by using <math>\mathbb N</math> or <math>\mathbb Z</math> (the set of all integers) with a subscript or superscript. Examples include <math>\mathbb{N}_1</math>,<ref name="Peano19012">{{cite book |last1=Peano |first1=Giuseppe |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |language=fr}}</ref> or <math>\mathbb Z^+</math><ref name="Grimaldi20042">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |year=2004 |isbn=978-0-201-72634-3 |edition=5th}}</ref> (for the set starting at 1) and <math>\mathbb{N}_0</math><ref>{{cite book |last1=Stewart |first1=Ian |url=https://books.google.com/books?id=xSN-BwAAQBAJ&dq=natural%20numbers%20%22N0%22&pg=PA160 |title=The Foundations of Mathematics |last2=Tall |first2=David |date=12 March 2015 |publisher=OUP Oxford |isbn=978-0-19-101648-6 |page=160 |language=en |access-date=30 July 2025}}</ref> or <math>\mathbb Z^{0+}</math><ref>{{cite book |last1=Fokas |first1=Athanassios |url=https://books.google.com/books?id=QwuhEAAAQBAJ&dq=natural%20numbers%20%22z0%2B%22&pg=PA4 |title=Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction |last2=Kaxiras |first2=Efthimios |date=12 December 2022 |publisher=World Scientific |isbn=978-1-80061-182-5 |page=4 |language=en |access-date=30 July 2025}}</ref> (for the set including 0).
+==Intuitive concept==
+An intuitive and implicit understanding of natural numbers is developed ''naturally'' through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the ''size of a collection;'' and ''a'' ''position in a sequence''.
+=== Size of a collection ===
+Natural numbers can be used to answer questions like: "how many apples are on the table?".<ref>{{Cite book |last1=Frege |first1=Gottlob |title=The foundations of arithmetic: a logico-mathematical enquiry into the concept of number |last2=Frege |first2=Gottlob |orig-date=1953|date=1975 |publisher=Northwestern Univ. Press |isbn=978-0-8101-0605-5 |edition=2. revised |location=Evanston Ill |pages=5}}</ref> A natural number used in this way describes a characteristic of a ''collection of objects''. This characteristic, the ''size of a collection'' is called [[cardinality]] and a natural number used to describe or measure it is called a cardinal number.
+[[File:Cardinality apples and oranges.svg|thumb|A group of apples and group of oranges with the same cardinality.]]
+Two collections have the same size or cardinality if there is a [[one-to-one correspondence]] between the objects in each collection to the objects in the other. For example, in the image to the right every apple can be paired off with one orange and every orange can be paired off with one apple. From this, even without counting or using numbers it can be seen that the group of apples has the ''same'' cardinality as the group of oranges, meaning they are both assigned the same cardinal number.
+The natural number 3 is the thing used for the particular cardinal number described above and for the cardinal number of any other collection of objects that could be paired off in the same way to one of these groups.
+=== Position in a sequence ===
+The natural numbers have a fixed progression, which is the familiar sequence beginning with 1, 2, 3, and so on. A natural number can be used to denote a specific position in any other sequence, in which case it is called an [[ordinal number|ordinal number.]] To have a specific position in a sequence means to come either before or after every other position in the sequence in a defined way, which is the concept of [[Order theory|order]].
+The natural number 3 then is the thing that ''comes after'' 2 and 1, and ''before'' 4, 5 and so on. The number 2 is the thing that ''comes after'' 1, and 1 is the first element in the sequence. Each number represents the relation that position bears to the rest of the infinite sequence.<ref>{{Cite journal |last=Benacerraf |first=Paul |date=January 1965 |title=What Numbers Could not Be |url=https://www.jstor.org/stable/2183530?origin=crossref |journal=The Philosophical Review |volume=74 |issue=1 |pages=47 |doi=10.2307/2183530 |url-access=subscription}}</ref>
+=== Counting ===
+The process of counting involves both the cardinal and ordinal use of the natural numbers and illustrates the way the two fit together. To count the number of objects in a collection, each object is paired off with a natural number, usually by mentally or verbally saying the name of the number and assigning it to a particular object. The numbers must be assigned in order starting with 1 (they are ordinal) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. When all of the objects have been assigned a number, the ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the whole collection.
 ==History==
@@ Line 33: / Line 58: @@
 [[Nicolas Chuquet]] used the term ''progression naturelle'' (natural progression) in 1484.<ref>{{cite book |last1=Chuquet |first1=Nicolas|author-link=Nicolas Chuquet |title=Le Triparty en la science des nombres|date=1881 |orig-date=1484 |url=https://gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image |language=fr}}</ref> The earliest known use of "natural number" as a complete English phrase is in 1763.<ref>{{cite book |last1=Emerson |first1=William |title=The method of increments|date=1763 |page=113 |url=https://archive.org/details/bim_eighteenth-century_the-method-of-increments_emerson-william_1763/page/112/mode/2up}}</ref><ref name="MacTutor"/> The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.<ref name="MacTutor">{{cite web |title=Earliest Known Uses of Some of the Words of Mathematics (N) |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/ |website=Maths History |language=en}}</ref>
-Starting at 0 or 1 has long been a matter of definition. In 1727, [[Bernard Le Bovier de Fontenelle]] wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |language=fr}}</ref> In 1889, [[Giuseppe Peano]] used N for the positive integers and started at 1,<ref>{{cite book |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up|page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref>{{cite book |last1=Peano |first1=Giuseppe |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up|language=fr}}</ref> Historically, most definitions have excluded 0,<ref name="MacTutor"/><ref>{{cite book |last1=Fine |first1=Henry Burchard |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |language=en}}</ref><ref>{{cite book |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |language=en}}</ref> but many mathematicians such as [[George A. Wentworth]], [[Bertrand Russell]], [[Nicolas Bourbaki]], [[Paul Halmos]], [[Stephen Cole Kleene]], and [[John Horton Conway]] have preferred to include 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor"/>
+Starting at 0 or 1 has long been a matter of definition. In 1727, [[Bernard Le Bovier de Fontenelle]] wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |language=fr}}</ref> In 1889, [[Giuseppe Peano]] used N for the positive integers and started at 1,<ref>{{cite book |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up|page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref name="Peano1901">{{cite book |last1=Peano |first1=Giuseppe |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up|language=fr}}</ref> Historically, most definitions have excluded 0,<ref name="MacTutor"/><ref>{{cite book |last1=Fine |first1=Henry Burchard |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |language=en}}</ref><ref>{{cite book |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |language=en}}</ref> but many mathematicians such as [[George A. Wentworth]], [[Bertrand Russell]], [[Nicolas Bourbaki]], [[Paul Halmos]], [[Stephen Cole Kleene]], and [[John Horton Conway]] have preferred to include 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor"/> This approach gained wider adoption in the 1960s<ref name="MacTutor"/> and was formalized in [[ISO 31-11]] (1978), which defines natural numbers to include zero, a convention retained in the current [[ISO/IEC 80000|ISO 80000-2]] standard.<ref name="ISO80000">{{cite book |url=https://www.iso.org/standard/64973.html |title=ISO 80000-2:2019 Quantities and units Part 2: Mathematics |date=24 June 2025 |publisher=[[International Organization for Standardization]] |chapter=Standard number sets and intervals |chapter-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10}}</ref>
-Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,<ref name="MacTutor"/>{{efn|name=MacLaneBirkhoff1999p15|{{harvtxt|Mac Lane|Birkhoff|1999|page=15}} include zero in the natural numbers: 'Intuitively, the set <math>\N=\{0,1,2,\ldots\}</math> of all ''natural numbers'' may be described as follows: <math>\N</math> contains an "initial" number {{math|0}}; ...'. They follow that with their version of the [[Peano's axioms]].}} number theory and analysis texts excluding 0,<ref name="MacTutor"/><ref name="Křížek">{{cite book |last1=Křížek |first1=Michal |last2=Somer |first2=Lawrence |last3=Šolcová |first3=Alena |title=From Great Discoveries in Number Theory to Applications |date=21 September 2021 |publisher=Springer Nature |isbn=978-3-030-83899-7 |page=6 |url=https://books.google.com/books?id=tklEEAAAQBAJ&dq=natural%20numbers%20zero&pg=PA6 |language=en}}</ref><ref>See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}}</ref> logic and set theory texts including 0,<ref>{{cite book |last1=Gowers |first1=Timothy |title=The Princeton companion to mathematics |date=2008 |publisher=Princeton university press |location=Princeton |isbn=978-0-691-11880-2 |page=17}}</ref><ref>{{cite book |last1=Bagaria |first1=Joan |title=Set Theory |url=http://plato.stanford.edu/entries/set-theory/ |publisher=The Stanford Encyclopedia of Philosophy |edition=Winter 2014 |year=2017 |access-date=13 February 2015 |archive-url=https://web.archive.org/web/20150314173026/http://plato.stanford.edu/entries/set-theory/ |archive-date=14 March 2015 |url-status=live}}</ref><ref>{{cite book |last1=Goldrei |first1=Derek |title=Classic Set Theory: A guided independent study |url=https://archive.org/details/classicsettheory00gold |url-access=limited |date=1998 |publisher=Chapman & Hall/CRC |location=Boca Raton, Fla. [u.a.] |isbn=978-0-412-60610-6 |page=[https://archive.org/details/classicsettheory00gold/page/n39 33] |edition=1. ed., 1. print|chapter=3}}</ref> dictionaries excluding 0,<ref name="MacTutor"/><ref>{{cite dictionary|url=http://www.merriam-webster.com/dictionary/natural%20number|title=natural number|dictionary=Merriam-Webster.com|publisher=[[Merriam-Webster]]|access-date=4 October 2014|archive-url=https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number| archive-date=13 December 2019|url-status=live}}</ref> school books (through high-school level) excluding 0, and upper-division college-level books including 0.<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include [[division by zero]]<ref name="Křížek"/> and the size of the [[empty set]]. [[Computer language]]s 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]].<ref>{{cite journal |last1=Brown |first1=Jim |title=In defense of index origin 0 |journal=ACM SIGAPL APL Quote Quad |date=1978 |volume=9 |issue=2 |page=7 |doi=10.1145/586050.586053|s2cid=40187000 }}</ref><ref>{{cite web |last1=Hui |first1=Roger |title=Is index origin 0 a hindrance? |url=http://www.jsoftware.com/papers/indexorigin.htm |website=jsoftware.com |access-date=19 January 2015 |archive-url=https://web.archive.org/web/20151020195547/http://www.jsoftware.com/papers/indexorigin.htm |archive-date=20 October 2015 |url-status=live}}</ref> Including 0 began to rise in popularity in the 1960s.<ref name="MacTutor"/> The [[ISO 31-11]] standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as [[ISO/IEC 80000|ISO 80000-2]].<ref name=ISO80000/>
 ===Formal construction===
 In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. [[Henri Poincaré]] stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<ref>{{cite book |last1=Poincaré |first1=Henri|translator1-first=William John|translator1-last= Greenstreet |title=La Science et l'hypothèse|trans-title=Science and Hypothesis|orig-date=1902|date=1905|chapter=On the nature of mathematical reasoning|chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1|at=VI}}</ref> [[Leopold Kronecker]] summarized his belief as "God made the integers, all else is the work of man".{{efn|The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<ref>{{cite book
-  |last=Gray |first=Jeremy |author-link=Jeremy Gray
+  |last=Gray |first=Jeremy |author-link=Jeremy Gray (mathematician)
   |year=2008
   |title=Plato's Ghost: The modernist transformation of mathematics
@@ Line 86: / Line 109: @@
   | year = 1993}}</ref> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include [[Goodstein's theorem]].<ref>{{cite journal | last1=Kirby | first1=Laurie | last2=Paris | first2=Jeff | title=Accessible Independence Results for Peano Arithmetic | journal=Bulletin of the London Mathematical Society | publisher=Wiley | volume=14 | issue=4 | year=1982 | issn=0024-6093 | doi=10.1112/blms/14.4.285 | pages=285–293}}</ref>
-==Notation==
+==Formal definitions==
-The [[Set (mathematics)|set]] of all natural numbers is standardly denoted {{math|'''N'''}} or <math>\mathbb N.</math><ref name=":1"/><ref>{{cite web |title=Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions |url=https://functions.wolfram.com/Notations/1/ |access-date=27 July 2020 |website=functions.wolfram.com}}</ref> Older texts have occasionally employed {{math|''J''}} as the symbol for this set.<ref>{{cite book |url=https://archive.org/details/1979RudinW |title=Principles of Mathematical Analysis |last=Rudin |first=W. |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8 |location=New York |page=25}}</ref>
+Formal definitions of the natural numbers take the existing, intuitive notion of natural numbers and the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. The two standard methods for doing this are: the [[Peano axioms]]; and [[set theory]].
+The Peano axioms (named for [[Giuseppe Peano]]) do not explicitly define what the natural numbers ''are'', but instead comprise a list of statements or [[Axiom|axioms]] that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particular [[Set (mathematics)|set]], in which a set can be generally understood as a collection of distinct objects or [[Element of a set|elements]]. While the two approaches are different, they are consistent in that the natural number sets collectively ''satisfy'' the Peano axioms.
+===Peano axioms===
+{{Main|Peano axioms}}
+The five Peano axioms are the following:<ref>{{cite encyclopedia
+ |editor-first=G.E. |editor-last=Mints
+ |title=Peano axioms
+ |encyclopedia=Encyclopedia of Mathematics
+ |publisher=[[Springer Science+Business Media|Springer]], in cooperation with the [[European Mathematical Society]]
+ |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms
+ |url-status=live |access-date=8 October 2014
+|archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms
+ |archive-date=13 October 2014
+}}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117&nbsp;ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/>
+{{harvtxt|Halmos|1974|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/>
+{{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')
+}}
+# 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 <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>.
+# 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 <math> x</math> is <math> x + 1</math>.
+===Set-theoretic definition===
+{{Main|Set-theoretic definition of natural numbers|von Neumann ordinal}}
+In set theory each natural number {{mvar|n}} is defined as an explicitly defined set, whose elements allow counting the elements of other sets. A variety of different constructions have been proposed, however the standard solution (due to [[John von Neumann]])<ref name="vonNeumann1923pp199-208">{{Harvp|von&nbsp;Neumann|1923}}</ref> is to define each natural number {{mvar|n}} as a set containing {{mvar|n}} elements in the following way:
+* Call {{math|0 {{=}} {{mset| }}}}, the [[empty set]].
+* Define the ''successor'' {{math|''S''(''a'')}} of any set {{mvar|a}} by {{math|''S''(''a'') {{=}} ''a'' ∪ {{mset|''a''}}}}.
+* By the [[axiom of infinity]], there exist sets which contain 0 and are [[closure (mathematics)|closed]] under the successor function. Such sets are said to be ''inductive''. The intersection of all inductive sets is still an inductive set.
+* This intersection is the set of the ''natural numbers''.
+This produces an iterative definition of the natural numbers satisfying the Peano axioms, sometimes called [[von Neumann ordinals]]:
-Since natural numbers may contain {{math|0}} or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:<ref name="ISO80000"/><ref name="Grimaldi">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |isbn=978-0-201-72634-3 |edition=5th |year=2004}}</ref>
+* {{math|0 {{=}} {{mset| }}}}
-* Naturals without zero: <math>\{1,2,...\}=\mathbb{N}^*= \mathbb N^+=\mathbb{N}_0\smallsetminus\{0\} = \mathbb{N}_1</math>
+* {{math|1 {{=}} 0 ∪ {{mset|0}} {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}}
-* Naturals with zero: <math>\;\{0,1,2,...\}=\mathbb{N}_0=\mathbb N^0=\mathbb{N}^*\cup\{0\}</math>
+* {{math|2 {{=}} 1 ∪ {{mset|1}} {{=}} {{mset|0, 1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}
+* {{math|3 {{=}} 2 ∪ {{mset|2}} {{=}} {{mset|0, 1, 2}}  {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}}
+* {{math|''n'' {{=}} ''n''−1 ∪ {{mset|''n''−1}} {{=}} {{mset|0, 1, ..., ''n''−1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, ..., {{mset|{{mset| }}, {{mset|{{mset| }}}}, ...}}}}}}
-Alternatively, since the natural numbers naturally form a [[subset]] of the [[integer]]s (often {{nowrap|denoted <math>\mathbb Z</math>),}} they may be referred to as the positive, or the non-negative integers, respectively.<ref>{{cite book |last1=Grimaldi |first1=Ralph P. |title=A review of discrete and combinatorial mathematics |date=2003 |publisher=Addison-Wesley |location=Boston |isbn=978-0-201-72634-3 |page=133 |edition=5th}}</ref> To be unambiguous about whether 0 is included or not, sometimes a superscript "<math>*</math>" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:<ref name="ISO80000">{{cite book |url=https://www.iso.org/standard/64973.html |title=ISO 80000-2:2019 Quantities and units Part 2: Mathematics |date=24 June 2025 |publisher=[[International Organization for Standardization]] |page=}}</ref>
+In this definition each natural number is equal to the set of all natural numbers less than it. Given a natural number {{math|''n''}}, the sentence "a set {{mvar|S}} has {{mvar|n}} elements" can be formally defined as "there exists a [[bijection]] from {{mvar|n}} to {{mvar|S}}." This formalizes the operation of ''counting'' the elements of {{mvar|S}}. Also, {{math|''n'' ≤ ''m''}} if and only if {{math|''n''}} is a [[subset]] of {{math|''m''}}. In other words, the [[set inclusion]] defines the usual [[total order]] on the natural numbers. This order is a [[well-order]].
-:<math>\{1, 2, 3,\dots\} = \{x \in \mathbb Z : x > 0\}=\mathbb Z^+= \mathbb{Z}_{>0}</math>
+Another construction sometimes called '''{{vanchor|Zermelo ordinals}}'''<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> defines {{math|0 {{=}} {{mset| }}}} and {{math|''S''(''a'') {{=}} {{mset|''a''}}}} and is now largely only of historical interest.
-:<math>\{0, 1, 2,\dots\} = \{x \in \mathbb Z : x \ge 0\}=\mathbb Z^{+}_{0}=\mathbb{Z}_ {\ge 0}</math>
 ==Properties==
@@ Line 140: / Line 203: @@
 ==Generalizations==
-Two important generalizations of natural numbers arise from the two uses of counting and ordering: [[cardinal number]]s and [[ordinal number]]s.
+Natural numbers are broadly used in two ways: to quantify and to order. A number used to represent the quantity of objects in a collection ("there are 6 coins on the table") is called a [[cardinal numeral]], while a number used to order individual objects within a collection ("she finished 6th in the race") is an [[ordinal numeral]].
-* 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. The numbering of cardinals usually begins at zero, to accommodate the [[empty set]] <math>\emptyset</math>. 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]] ({{math|{{not a typo|ℵ}}<sub>0</sub>}}).
-* Natural numbers are also used as [[Ordinal numbers (linguistics)|linguistic ordinal numbers]]: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the  [[empty set]] <math>\emptyset</math>. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any [[well-order]]ed countably infinite set without [[limit points]]. 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 {{math|ω}}; this is also the ordinal number of the set of natural numbers itself.
-The least ordinal of cardinality {{math|{{not a typo|ℵ}}<sub>0</sub>}} (that is, the [[Von Neumann cardinal assignment|initial ordinal]] of {{math|{{not a typo|ℵ}}<sub>0</sub>}}) is {{math|ω}} but many well-ordered sets with cardinal number {{math|{{not a typo|ℵ}}<sub>0</sub>}} have an ordinal number greater than {{math|ω}}.
-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]]. Other generalizations are discussed in {{section link|Number#Extensions of the concept}}.
-[[Georges Reeb]] used to claim provocatively that "The naïve integers don't fill up <math>\mathbb{N}</math>".<ref>{{cite journal |title=Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others |journal=Real Analysis Exchange |date=2017 |volume=42 |issue=2 |pages=193–253 |doi=10.14321/realanalexch.42.2.0193|doi-access=free|arxiv=1703.00425 |last1=Fletcher |first1=Peter |last2=Hrbacek |first2=Karel |last3=Kanovei |first3=Vladimir |last4=Katz |first4=Mikhail G. |last5=Lobry |first5=Claude |last6=Sanders |first6=Sam }}</ref>
-==Formal definitions==
-There are two standard methods for formally defining natural numbers. The first one, named for [[Giuseppe Peano]], consists of an autonomous [[axiomatic theory]] called [[Peano arithmetic]], based on few axioms called [[Peano axioms]].
-The second definition is based on [[set theory]]. It defines the natural numbers as specific [[set (mathematics)|set]]s. More precisely, each natural number {{mvar|n}} is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set {{mvar|S}} has {{mvar|n}} elements" means that there exists a [[one to one correspondence]] between the two sets {{mvar|n}} and {{mvar|S}}.
-The sets used to define natural numbers satisfy Peano axioms. It follows that every [[theorem]] that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not ''provable'' inside Peano arithmetic. A probable example is [[Fermat's Last Theorem]].
-The definition of the integers as sets satisfying Peano axioms provide a [[model (mathematical logic)|model]] of Peano arithmetic inside set theory. An important consequence is that, if set theory is [[consistent]] (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong.
-===Peano axioms===
-{{Main|Peano axioms}}
-The five Peano axioms are the following:<ref>{{cite encyclopedia
- |editor-first=G.E. |editor-last=Mints
- |title=Peano axioms
- |encyclopedia=Encyclopedia of Mathematics
- |publisher=[[Springer Science+Business Media|Springer]], in cooperation with the [[European Mathematical Society]]
- |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms
- |url-status=live |access-date=8 October 2014
-|archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms
- |archive-date=13 October 2014
-}}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117&nbsp;ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/>
-{{harvtxt|Halmos|1960|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/>
-{{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')
-}}
-# 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 <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>.
-# 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 <math> x</math> is <math> x + 1</math>.
-===Set-theoretic definition===
-{{Main|Set-theoretic definition of natural numbers|von Neumann ordinal}}
-Intuitively, the natural number {{mvar|n}} is the common property of all [[set (mathematics)|set]]s that have {{mvar|n}} elements. So, it seems natural to define {{mvar|n}} as an [[equivalence class]] under the relation "can be made in [[one to one correspondence]]". This does not work in all [[set theory|set theories]], as such an equivalence class would not be a set{{efn|In some set theories, e.g., [[New Foundations]], a [[universal set]] exists and Russel's paradox cannot be formulated.}} (because of [[Russell's paradox]]). The standard solution is to define a particular set with {{mvar|n}} elements that will be called the natural number {{mvar|n}}.
-The following definition was first published by [[John von Neumann]],<ref name="vonNeumann1923pp199-208">{{Harvp|von&nbsp;Neumann|1923}}</ref> although Levy attributes the idea to unpublished work of Zermelo in 1916.<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> As this definition extends to [[infinite set]] as a definition of [[ordinal number]], the sets considered below are sometimes called [[von Neumann ordinals]].
-The definition proceeds as follows:
-* Call {{math|0 {{=}} {{mset| }}}}, the [[empty set]].
-* Define the ''successor'' {{math|''S''(''a'')}} of any set {{mvar|a}} by {{math|''S''(''a'') {{=}} ''a'' ∪ {{mset|''a''}}}}.
-* By the [[axiom of infinity]], there exist sets which contain 0 and are [[closure (mathematics)|closed]] under the successor function. Such sets are said to be ''inductive''. The intersection of all inductive sets is still an inductive set.
-* This intersection is the set of the ''natural numbers''.
-It follows that the natural numbers are defined iteratively as follows:
-:*{{math|0 {{=}} {{mset| }}}},
-:*{{math|1 {{=}} 0 ∪ {{mset|0}} {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}},
-:*{{math|2 {{=}} 1 ∪ {{mset|1}} {{=}} {{mset|0, 1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}},
-:*{{math|3 {{=}} 2 ∪ {{mset|2}} {{=}} {{mset|0, 1, 2}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}},
-:*{{math|''n'' {{=}} ''n''−1 ∪ {{mset|''n''−1}} {{=}} {{mset|0, 1, ..., ''n''−1}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, ..., {{mset|{{mset| }}, {{mset|{{mset| }}}}, ...}}}}}},
-:* etc.
-It can be checked that the natural numbers satisfy the [[Peano axioms]].
-With this definition, given a natural number {{math|''n''}}, the sentence "a set {{mvar|S}} has {{mvar|n}} elements" can be formally defined as "there exists a [[bijection]] from {{mvar|n}} to {{mvar|S}}." This formalizes the operation of ''counting'' the elements of {{mvar|S}}. Also, {{math|''n'' ≤ ''m''}} if and only if {{math|''n''}} is a [[subset]] of {{math|''m''}}. In other words, the [[set inclusion]] defines the usual [[total order]] on the natural numbers. This order is a [[well-order]].
-It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the [[von Neumann ordinal|von Neumann definition of ordinals]] for defining all [[ordinal number]]s, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
-If one [[finitism|does not accept the axiom of infinity]], the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
-There are other set theoretical constructions. In particular, [[Ernst Zermelo]] provided a construction that is nowadays only of historical interest, and is sometimes referred to as '''{{vanchor|Zermelo ordinals}}'''.<ref name="Levy"/> It consists in defining {{math|0}} as the empty set, and {{math|''S''(''a'') {{=}} {{mset|''a''}}}}.
+These two uses of natural numbers apply only to [[finite set]]s. [[Georg Cantor]] discovered at the end of the 19th century that both uses of natural numbers can be generalized to [[infinite set]]s, but that they lead to two different concepts of "infinite" numbers, the [[cardinal number]]s and the [[ordinal number]]s.
-With this definition each nonzero natural number is a [[singleton set]]. So, the property of the natural numbers to represent [[cardinalities]] is not directly accessible; only the ordinal property (being the {{mvar|n}}th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
+Other generalizations of natural numbers are discussed in {{section link|Number#Extensions of the concept}}.
 ==See also==
@@ Line 272: / Line 261: @@
   |title=Essays on the Theory of Numbers
   |publisher=Dover Books
-  |isbn=978-0-486-21010-0
+  |oclc=552490
   |via=Archive.org
   |url=https://archive.org/details/essaysontheoryof0000dede
@@ Line 305: / Line 294: @@
 * {{cite book
   |last=Halmos |first=Paul |author-link=Paul Halmos
-  |year=1960
+  |year=1974
   |title=Naive Set Theory
   |publisher=Springer Science & Business Media

Natural number: Difference between revisions

Latest revision as of 11:58, 3 November 2025

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