Natural number

Inmathematics,thenatural numbersare thenumbers0, 1, 2, 3, and so on, possibly excluding 0.^[1]Some start counting with 0, defining the natural numbers as thenon-negative integers $0, 1, 2, 3,...$ ,while others start with 1, defining them as thepositive integers $1, 2, 3,...$ .^[a]Some authors acknowledge both definitions whenever convenient.^[2]Sometimes, thewhole numbersare the natural numbers plus zero. In other cases, thewhole numbersrefer to all of theintegers,including negative integers.^[3]Thecounting numbersare another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1.^[4]

The natural numbers are used for counting things, like "there aresixcoins on the table ", in which case they are calledcardinal numbers.They are also used to put things in order, like "this is thethirdlargest city in the country ", which are calledordinal numbers.Natural numbers are also used as labels, likejersey numberson a sports team, where they serve asnominal numbersand do not have mathematical properties.^[2]^[5]

The natural numbers form aset,commonly symbolized as a bold $N$ orblackboard bold⁠ $\mathbb {N}$ ⁠.Many othernumber setsare built from the natural numbers. For example, theintegersare made by adding 0 and negative numbers. Therational numbersadd fractions, and thereal numbersadd infinite decimals.Complex numbersadd thesquare root of $-1$ .This chain of extensions canonicallyembedsthe natural numbers in the other number systems.^[6]^[7]

Natural numbers are studied in different areas of math.Number theorylooks at things like how numbers divide evenly (divisibility), or howprime numbersare spread out.Combinatoricsstudies counting and arranging numbered objects, such aspartitionsandenumerations.

History

Ancient roots

The most primitive method of representing a natural number is to use one's fingers, as infinger counting.Putting down atally markfor 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 ofnumeralsto represent numbers. This allowed systems to be developed for recording large numbers. The ancientEgyptiansdeveloped a powerful system of numerals with distincthieroglyphsfor 1, 10, and all powers of 10 up to over 1 million. A stone carving fromKarnak,dating back from around 1500 BCE and now at theLouvrein Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. TheBabylonianshad aplace-valuesystem 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.^[11]

A much later advance was the development of the idea that0can be considered as a number, with its own numeral. The use of a 0digitin 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.^[b]TheOlmecandMaya civilizationsused 0 as a separate number as early as the1st century BCE,but this usage did not spread beyondMesoamerica.^[13]^[14]The use of a numeral 0 in modern times originated with the Indian mathematicianBrahmaguptain 628 CE. However, 0 had been used as a number in the medievalcomputus(the calculation of the date of Easter), beginning withDionysius Exiguusin 525 CE, without being denoted by a numeral. StandardRoman numeralsdo not have a symbol for 0; instead,nulla(or the genitive formnullae) fromnullus,the Latin word for "none", was employed to denote a 0 value.^[15]

The first systematic study of numbers asabstractionsis usually credited to theGreekphilosophersPythagorasandArchimedes.Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.^[c]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).^[17]However, in the definition ofperfect numberwhich comes shortly afterward, Euclid treats 1 as a number like any other.^[18]

Independent studies on numbers also occurred at around the same time inIndia,China, andMesoamerica.^[19]

Emergence as a term

Nicolas Chuquetused the termprogression naturelle(natural progression) in 1484.^[20]The earliest known use of "natural number" as a complete English phrase is in 1763.^[21]^[22]The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.^[22]

Starting at 0 or 1 has long been a matter of definition. In 1727,Bernard Le Bovier de Fontenellewrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.^[23]In 1889,Giuseppe Peanoused N for the positive integers and started at 1,^[24]but he later changed to using N₀and N₁.^[25]Historically, most definitions have excluded 0,^[22]^[26]^[27]but many mathematicians such asGeorge A. Wentworth,Bertrand Russell,Nicolas Bourbaki,Paul Halmos,Stephen Cole Kleene,andJohn Horton Conwayhave preferred to include 0.^[28]^[22]

Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,^[22]^[d]number theory and analysis texts excluding 0,^[22]^[29]^[30]logic and set theory texts including 0,^[31]^[32]^[33]dictionaries excluding 0,^[22]^[34]school books (through high-school level) excluding 0, and upper-division college-level books including 0.^[1]There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised includedivision by zero^[29]and the size of theempty set.Computer languagesoftenstart from zerowhen enumerating items likeloop countersandstring-orarray-elements.^[35]^[36]Including 0 began to rise in popularity in the 1960s.^[22]TheISO 31-11standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition asISO 80000-2.^[37]

Formal construction

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.Henri Poincaréstated 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 Kroneckersummarized his belief as "God made the integers, all else is the work of man".^[e]

Theconstructivistssaw a need to improve upon the logical rigor in thefoundations of mathematics.^[f]In the 1860s,Hermann Grassmannsuggested arecursive definitionfor 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 numberswere initiated byFrege.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, includingRussell'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.^[41]

In 1881,Charles Sanders Peirceprovided the firstaxiomatizationof natural-number arithmetic.^[42]^[43]In 1888,Richard Dedekindproposed another axiomatization of natural-number arithmetic,^[44]and in 1889, Peano published a simplified version of Dedekind's axioms in his bookThe principles of arithmetic presented by a new method(Latin:Arithmetices principia, nova methodo exposita). This approach is now calledPeano arithmetic.It is based on anaxiomatizationof the properties ofordinal numbers:each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic isequiconsistentwith several weak systems ofset theory.One such system isZFCwith theaxiom of infinityreplaced by its negation.^[45]Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms includeGoodstein's theorem.^[46]

Notation

Thesetof all natural numbers is standardly denoted $N$ or $\mathbb {N}.$ ^[2]^[47]Older texts have occasionally employed $J$ as the symbol for this set.^[48]

Since natural numbers may contain $0$ 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:^[37]^[49]

Naturals without zero: $\{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}$
Naturals with zero: $\;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}$

Alternatively, since the natural numbers naturally form asubsetof theintegers(oftendenoted $\mathbb {Z}$ ),they may be referred to as the positive, or the non-negative integers, respectively.^[50]To be unambiguous about whether 0 is included or not, sometimes a superscript " $*$ "or" + "is added in the former case, and a subscript (or superscript)" 0 "is added in the latter case:^[37]

\{1,2,3,\dots \}=\{x\in \mathbb {Z}:x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}

\{0,1,2,\dots \}=\{x\in \mathbb {Z}:x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}

Properties

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

Addition

Given the set $\mathbb {N}$ of natural numbers and thesuccessor function $S\colon \mathbb {N} \to \mathbb {N}$ sending each natural number to the next one, one can defineadditionof natural numbers recursively by setting $a + 0 = a$ and $a + S (b) = S (a + b)$ for all $a$ , $b$ .Thus, $a + 1 = a + S(0) = S(a +0) = S(a)$ , $a + 2 = a + S(1) = S(a +1) = S(S(a))$ ,and so on. Thealgebraic structure $(\mathbb {N},+)$ is acommutative monoidwithidentity element0. It is afree monoidon one generator. This commutative monoid satisfies thecancellation property,so it can be embedded in agroup.The smallest group containing the natural numbers is theintegers.

If 1 is defined as $S (0)$ ,then $b + 1 = b + S (0) = S (b + 0) = S (b)$ .That is, $b + 1$ is simply the successor of $b$ .

Multiplication

Analogously, given that addition has been defined, amultiplicationoperator $\times$ can be defined via $a \times 0 = 0$ and $a \times S(b) = (a \times b) + a$ .This turns $(\mathbb {N} ^{*},\times )$ into afree commutative monoidwith identity element 1; a generator set for this monoid is the set ofprime numbers.

Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in thedistribution law: $a \times (b + c) = (a \times b) + (a \times c)$ .These properties of addition and multiplication make the natural numbers an instance of acommutative 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 $\mathbb {N}$ is notclosedunder subtraction (that is, subtracting one natural from another does not always result in another natural), means that $\mathbb {N}$ isnotaring;instead it is asemiring(also known as arig).

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 $a + 1 = S (a)$ and $a \times 1 = a$ .Furthermore, $(\mathbb {N^{*}},+)$ has no identity element.

Order

In this section, juxtaposed variables such as $ab$ indicate the product $a \times b$ ,^[51]and the standardorder of operationsis assumed.

Atotal orderon the natural numbers is defined by letting $a \leq b$ if and only if there exists another natural number $c$ where $a + c = b$ .This order is compatible with thearithmetical operationsin the following sense: if $a$ , $b$ and $c$ are natural numbers and $a \leq b$ ,then $a + c \leq b + c$ and $ac \leq bc$ .

An important property of the natural numbers is that they arewell-ordered:every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by anordinal number;for the natural numbers, this is denoted as $ω$ (omega).

Division

In this section, juxtaposed variables such as $ab$ indicate the product $a \times b$ ,and the standardorder of operationsis assumed.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure ofdivision with remainderorEuclidean divisionis available as a substitute: for any two natural numbers $a$ and $b$ with $b \neq 0$ there are natural numbers $q$ and $r$ such that

a=bq+r{\text{ and }}r<b.

The number $q$ is called thequotientand $r$ is called theremainderof the division of $a$ by $b$ .The numbers $q$ and $r$ are uniquely determined by $a$ and $b$ .This Euclidean division is key to the several other properties (divisibility), algorithms (such as theEuclidean 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:

Closureunder addition and multiplication: for all natural numbers $a$ and $b$ ,both $a + b$ and $a \times b$ are natural numbers.^[52]
Associativity:for all natural numbers $a$ , $b$ ,and $c$ , $a + (b + c) = (a + b) + c$ and $a \times (b \times c) = (a \times b) \times c$ .^[53]
Commutativity:for all natural numbers $a$ and $b$ , $a + b = b + a$ and $a \times b = b \times a$ .^[54]
Existence ofidentity elements:for every natural numbera,a+ 0 =aanda× 1 =a.
- If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number $a$ , $a \times 1 = a$ .However, the "existence of additive identity element" property is not satisfied
Distributivityof multiplication over addition for all natural numbers $a$ , $b$ ,and $c$ , $a \times (b + c) = (a \times b) + (a \times c)$ .
No nonzerozero divisors:if $a$ and $b$ are natural numbers such that $a \times b = 0$ ,then $a = 0$ or $b = 0$ (or both).

Generalizations

Two important generalizations of natural numbers arise from the two uses of counting and ordering:cardinal numbersandordinal 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. The numbering of cardinals usually begins at zero, to accommodate theempty set $\emptyset$ .This concept of "size" relies on maps between sets, such that two sets havethe same size,exactly if there exists abijectionbetween them. The set of natural numbers itself, and any bijective image of it, is said to becountably infiniteand to havecardinality aleph-null( $ℵ 0$ ).
Natural numbers are also used aslinguistic ordinal numbers:"first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of theempty set $\emptyset$ .This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of anywell-orderedcountably infinite set withoutlimit 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 anorder 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 $ω$ ;this is also the ordinal number of the set of natural numbers itself.

The least ordinal of cardinality $ℵ 0$ (that is, theinitial ordinalof $ℵ 0$ ) is $ω$ but many well-ordered sets with cardinal number $ℵ 0$ have an ordinal number greater than $ω$ .

Forfinitewell-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 countablenon-standard model of arithmeticsatisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed bySkolemin 1933. Thehypernaturalnumbers are an uncountable model that can be constructed from the ordinary natural numbers via theultrapower construction.Other generalizations are discussed inNumber § Extensions of the concept.

Georges Reebused to claim provocatively that "The naïve integers don't fill up $\mathbb {N}$ ".^[55]

Formal definitions

There are two standard methods for formally defining natural numbers. The first one, named forGiuseppe Peano,consists of an autonomousaxiomatic theorycalledPeano arithmetic,based on few axioms calledPeano axioms.

The second definition is based onset theory.It defines the natural numbers as specificsets.More precisely, each natural number $n$ is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set $S$ has $n$ elements "means that there exists aone to one correspondencebetween the two sets $n$ and $S$ .

The sets used to define natural numbers satisfy Peano axioms. It follows that everytheoremthat 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 notprovableinside Peano arithmetic. A probable example isFermat's Last Theorem.

The definition of the integers as sets satisfying Peano axioms provide amodelof Peano arithmetic inside set theory. An important consequence is that, if set theory isconsistent(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

The five Peano axioms are the following:^[56]^[g]

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 $y$ .
Theaxiom 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

Intuitively, the natural number $n$ is the common property of allsetsthat have $n$ elements. So, it seems natural to define $n$ as anequivalence classunder the relation "can be made inone to one correspondence".This does not work in allset theories,as such an equivalence class would not be a set^[h](because ofRussell's paradox). The standard solution is to define a particular set with $n$ elements that will be called the natural number $n$ .

The following definition was first published byJohn von Neumann,^[57]although Levy attributes the idea to unpublished work of Zermelo in 1916.^[58]As this definition extends toinfinite setas a definition ofordinal number,the sets considered below are sometimes calledvon Neumann ordinals.

The definition proceeds as follows:

Call $0 = { }$ ,theempty set.
Define thesuccessor $S (a)$ of any set $a$ by $S (a) = a \cup {a}$ .
By theaxiom of infinity,there exist sets which contain 0 and areclosedunder the successor function. Such sets are said to beinductive.The intersection of all inductive sets is still an inductive set.
This intersection is the set of thenatural numbers.

It follows that the natural numbers are defined iteratively as follows:

$0 = { }$ ,
$1 = 0 \cup {0} = {0} = {{ }}$ ,
$2 = 1 \cup {1} = {0, 1} = {{ }, {{ }}}$ ,
$3 = 2 \cup {2} = {0, 1, 2}$ $= {{ }, {{ }}, {{ }, {{ }}}}$ ,
$n = n -1 \cup {n -1} = {0, 1,..., n -1}$ $= {{ }, {{ }},..., {{ }, {{ }},...}}$ ,
etc.

It can be checked that the natural numbers satisfy thePeano axioms.

With this definition, given a natural number $n$ ,the sentence "a set $S$ has $n$ elements "can be formally defined as" there exists abijectionfrom $n$ to $S$ .This formalizes the operation ofcountingthe elements of $S$ .Also, $n \leq m$ if and only if $n$ is asubsetof $m$ .In other words, theset inclusiondefines the usualtotal orderon the natural numbers. This order is awell-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 thevon Neumann definition of ordinalsfor defining allordinal numbers,including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If onedoes 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 Zermeloprovided a construction that is nowadays only of historical interest, and is sometimes referred to asZermelo ordinals.^[58]It consists in defining $0$ as the empty set, and $S (a) = {a}$ .

With this definition each nonzero natural number is asingleton set.So, the property of the natural numbers to representcardinalitiesis not directly accessible; only the ordinal property (being the $n$ th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

Notes

^See§ Emergence as a term
^A tablet found at Kish... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.^[12]
^This convention is used, for example, inEuclid's Elements,see D. Joyce's web edition of Book VII.^[16]
^Mac Lane & Birkhoff (1999,p. 15) include zero in the natural numbers: 'Intuitively, the set $\mathbb {N} =\{0,1,2,\ldots \}$ of allnatural numbersmay be described as follows: $\mathbb {N}$ contains an "initial" number $0$ ;...'. They follow that with their version of thePeano's axioms.
^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."^[39]^[40]
^"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990,p. 606)
^Hamilton (1988,pp. 117 ff) calls them "Peano's Postulates" and begins with "1.0 is a natural number. "
Halmos (1960,p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ "( $ω$ is the set of all natural numbers).
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)
^In some set theories, e.g.,New Foundations,auniversal setexists and Russel's paradox cannot be formulated.

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^"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions".functions.wolfram.com.Retrieved27 July2020.
^Rudin, W. (1976).Principles of Mathematical Analysis.New York: McGraw-Hill. p. 25.ISBN 978-0-07-054235-8.
^Grimaldi, Ralph P. (2004).Discrete and Combinatorial Mathematics: An applied introduction(5th ed.). Pearson Addison Wesley.ISBN 978-0-201-72634-3.
^Grimaldi, Ralph P. (2003).A review of discrete and combinatorial mathematics(5th ed.). Boston: Addison-Wesley. p. 133.ISBN 978-0-201-72634-3.
^Weisstein, Eric W."Multiplication".mathworld.wolfram.com.Retrieved27 July2020.
^Fletcher, Harold; Howell, Arnold A. (9 May 2014).Mathematics with Understanding.Elsevier. p. 116.ISBN 978-1-4832-8079-0....the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
^Davisson, Schuyler Colfax (1910).College Algebra.Macmillian Company. p. 2.Addition of natural numbers is associative.
^Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).Laidlaw mathematics series.Vol. 8. Laidlaw Bros. p. 25.
^Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017)."Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others".Real Analysis Exchange.42(2): 193–253.arXiv:1703.00425.doi:10.14321/realanalexch.42.2.0193.
^Mints, G.E. (ed.)."Peano axioms".Encyclopedia of Mathematics.Springer,in cooperation with theEuropean Mathematical Society.Archivedfrom the original on 13 October 2014.Retrieved8 October2014.
^von Neumann (1923)
^^a ^bLevy (1979),p. 52

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External links

"Natural number",Encyclopedia of Mathematics,EMS Press,2001 [1994]
"Axioms and construction of natural numbers".apronus.com.

[2] See§ Emergence as a term

[14] A tablet found at Kish... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.^[12]

[19] This convention is used, for example, inEuclid's Elements,see D. Joyce's web edition of Book VII.^[16]

[MacLaneBirkhoff1999p15-32] Mac Lane & Birkhoff (1999,p. 15) include zero in the natural numbers: 'Intuitively, the set $\mathbb {N} =\{0,1,2,\ldots \}$ of allnatural numbersmay be described as follows: $\mathbb {N}$ contains an "initial" number $0$ ;...'. They follow that with their version of thePeano's axioms.

[45] 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."^[39]^[40]

[46] "Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990,p. 606)

[63] Hamilton (1988,pp. 117 ff) calls them "Peano's Postulates" and begins with "1.0 is a natural number. "
Halmos (1960,p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ "( $ω$ is the set of all natural numbers).
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)

[64] In some set theories, e.g.,New Foundations,auniversal setexists and Russel's paradox cannot be formulated.

[Enderton-1] Enderton, Herbert B. (1977).Elements of set theory.New York: Academic Press. p. 66.ISBN 0122384407.

[:1-3] Weisstein, Eric W."Natural Number".mathworld.wolfram.com.Retrieved11 August2020.

[4] Ganssle, Jack G. & Barr, Michael (2003)."integer".Embedded Systems Dictionary.Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer).ISBN 978-1-57820-120-4.Archivedfrom the original on 29 March 2017.Retrieved28 March2017– via Google Books.

[MathWorld_CountingNumber-5] Weisstein, Eric W."Counting Number".MathWorld.

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[7] Mendelson (2008,p. 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."

[8] Bluman (2010,p. 1): "Numbers make up the foundation of mathematics."

[RBINS_intro-9] "Introduction".Ishango bone.Brussels, Belgium:Royal Belgian Institute of Natural Sciences.Archived fromthe originalon 4 March 2016.

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[UNESCO-11] "The Ishango Bone, Democratic Republic of the Congo".UNESCO's Portal to the Heritage of Astronomy.Archived fromthe originalon 10 November 2014.,on permanent display at theRoyal Belgian Institute of Natural Sciences,Brussels, Belgium.

[12] Ifrah, Georges (2000).The Universal History of Numbers.Wiley.ISBN 0-471-37568-3.

[13] "A history of Zero".MacTutor History of Mathematics.Archivedfrom the original on 19 January 2013.Retrieved23 January2013.

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[Mueller_2006_p._58-20] Mueller, Ian (2006).Philosophy of mathematics and deductive structure inEuclid's Elements.Mineola, New York: Dover Publications. p. 58.ISBN 978-0-486-45300-2.OCLC 69792712.

[21] Euclid."Book VII, definition 22".In Joyce, D. (ed.).Elements.Clark University.A perfect number is that which is equal to the sum of its own parts.In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example $6 = 1 + 2 + 3$ is a perfect number.

[22] Kline, Morris (1990) [1972].Mathematical Thought from Ancient to Modern Times.Oxford University Press.ISBN 0-19-506135-7.

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[27] Arithmetices principia: nova methodo(in Latin). Fratres Bocca. 1889. p. 12.

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[29] Fine, Henry Burchard (1904).A College Algebra.Ginn. p. 6.

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[Křížek-33] Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021).From Great Discoveries in Number Theory to Applications.Springer Nature. p. 6.ISBN 978-3-030-83899-7.

[34] See, for example,Carothers (2000,p. 3) orThomson, Bruckner & Bruckner (2008,p. 2)

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[37] Goldrei, Derek (1998). "3".Classic Set Theory: A guided independent study(1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p.33.ISBN 978-0-412-60610-6.

[38] "natural number".Merriam-Webster.com.Merriam-Webster.Archivedfrom the original on 13 December 2019.Retrieved4 October2014.

[39] Brown, Jim (1978). "In defense of index origin 0".ACM SIGAPL APL Quote Quad.9(2): 7.doi:10.1145/586050.586053.S2CID 40187000.

[40] Hui, Roger."Is index origin 0 a hindrance?".jsoftware.com.Archivedfrom the original on 20 October 2015.Retrieved19 January2015.

[ISO80000-41] "Standard number sets and intervals"(PDF).ISO 80000-2:2019.International Organization for Standardization.19 May 2020. p. 4.

[42] Poincaré, Henri (1905) [1902]."On the nature of mathematical reasoning".La Science et l'hypothèse[Science and Hypothesis]. Translated by Greenstreet, William John. VI.

[43] Gray, Jeremy(2008).Plato's Ghost: The modernist transformation of mathematics.Princeton University Press. p. 153.ISBN 978-1-4008-2904-0.Archivedfrom the original on 29 March 2017 – via Google Books.

[44] Weber, Heinrich L. (1891–1892)."Kronecker".Jahresbericht der Deutschen Mathematiker-Vereinigung[Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived fromthe originalon 9 August 2018;"access toJahresbericht der Deutschen Mathematiker-Vereinigung".Archived fromthe originalon 20 August 2017.

[47] Eves 1990,Chapter 15

[48] Peirce, C. S.(1881)."On the Logic of Number".American Journal of Mathematics.4(1): 85–95.doi:10.2307/2369151.JSTOR 2369151.MR 1507856.

[49] Shields, Paul (1997)."3. Peirce's Axiomatization of Arithmetic".In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).Studies in the Logic of Charles Sanders Peirce.Indiana University Press. pp. 43–52.ISBN 0-253-33020-3.

[50] Was sind und was sollen die Zahlen?(in German). F. Vieweg. 1893. 71-73.

[51] Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity".Mathematical Logic Quarterly.39(3): 338–352.doi:10.1002/malq.19930390138.MR 1270381.

[52] Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic".Bulletin of the London Mathematical Society.14(4). Wiley: 285–293.doi:10.1112/blms/14.4.285.ISSN 0024-6093.

[53] "Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions".functions.wolfram.com.Retrieved27 July2020.

[54] Rudin, W. (1976).Principles of Mathematical Analysis.New York: McGraw-Hill. p. 25.ISBN 978-0-07-054235-8.

[Grimaldi-55] Grimaldi, Ralph P. (2004).Discrete and Combinatorial Mathematics: An applied introduction(5th ed.). Pearson Addison Wesley.ISBN 978-0-201-72634-3.

[56] Grimaldi, Ralph P. (2003).A review of discrete and combinatorial mathematics(5th ed.). Boston: Addison-Wesley. p. 133.ISBN 978-0-201-72634-3.

[57] Weisstein, Eric W."Multiplication".mathworld.wolfram.com.Retrieved27 July2020.

[58] Fletcher, Harold; Howell, Arnold A. (9 May 2014).Mathematics with Understanding.Elsevier. p. 116.ISBN 978-1-4832-8079-0....the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication

[59] Davisson, Schuyler Colfax (1910).College Algebra.Macmillian Company. p. 2.Addition of natural numbers is associative.

[60] Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).Laidlaw mathematics series.Vol. 8. Laidlaw Bros. p. 25.

[61] Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017)."Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others".Real Analysis Exchange.42(2): 193–253.arXiv:1703.00425.doi:10.14321/realanalexch.42.2.0193.

[62] Mints, G.E. (ed.)."Peano axioms".Encyclopedia of Mathematics.Springer,in cooperation with theEuropean Mathematical Society.Archivedfrom the original on 13 October 2014.Retrieved8 October2014.

[vonNeumann1923pp199-208-65] von Neumann (1923)

[Levy-66] Levy (1979),p. 52

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v t e Numbersystems
Sets ofdefinable numbers	Natural numbers( $\mathbb {N}$ ) Integers( $\mathbb {Z}$ ) Rational numbers( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers( $\mathbb {A}$ ) Closed-form numbers Periods( ${\mathcal {P}}$ ) Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers Gaussian rationals Eisenstein integers
Composition algebras	Division algebras:Real numbers( $\mathbb {R}$ ) Complex numbers( $\mathbb {C}$ ) Quaternions( $\mathbb {H}$ ) Octonions( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Otherhypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions( $\mathbb {S}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Infinitiesandinfinitesimals	Cardinal numbers Extended natural numbers Extended real numbers Projective Extended complex numbers Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers
Other types	Irrational numbers Fuzzy numbers Transcendental numbers p-adicnumbers(p-adicsolenoids) Profinite integers Normal numbers
Classification List