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Natural number

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Number used for counting

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, ...

Inmathematics, thenatural numbers are the numbers0,1,2,3, and so on, possibly excluding 0.[a][1] The termspositive integers,non-negative integers,whole numbers, andcounting numbers are also used.[2][3] Theset of the natural numbers is commonly denoted by a boldN or ablackboard bold⁠N{\displaystyle \mathbb {N} }⁠.

The natural numbers are used for counting, and for labeling the result of a count, such as: "there areseven days in a week", in which case they are calledcardinal numbers. They are also used to label places in an ordered series, such as: "thethird day of the month", in which case they are calledordinal numbers. Natural numbers can also be used to label, like thejersey numbers of a sports team; in this case, they have no specific mathematical properties and are callednominal numbers.[4]

Natural numbers can becompared by magnitude, with larger numbers coming after smaller ones in the list 1, 2, 3, .... Two basicarithmetical operations are defined on natural numbers:addition andmultiplication. However, the inverse operations,subtraction anddivision, only sometimes give natural-number results: subtracting a larger natural number from a smaller one results in anegative number and dividing one natural number by another commonly leaves aremainder.

The most commonnumber systems used throughout mathematics – theintegers,rational numbers,real numbers, andcomplex numbers – contain the natural numbers, and can be formally defined in terms of natural numbers.[5][6]

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

Terminology and notation

The termnatural numbers has two common definitions: either0, 1, 2, ... or1, 2, 3, .... Because there is no universal convention, the definition can be chosen to suit the context of use.[1][7] To eliminate ambiguity, the sequences1, 2, 3, ... and0, 1, 2, ... are often called thepositive integers and thenon-negative integers, respectively.

The phrasewhole 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]

Theset of all natural numbers is typically denotedN or inblackboard bold asN.{\displaystyle \mathbb {N} .}[7][11][b] Whether 0 is included is often determined by the context but may also be specified by usingN{\displaystyle \mathbb {N} } orZ{\displaystyle \mathbb {Z} } (the set of all integers) with a subscript or superscript. Examples includeN1{\displaystyle \mathbb {N} _{1}},[13] orZ+{\displaystyle \mathbb {Z} ^{+}}[14] (for the set starting at 1) andN0{\displaystyle \mathbb {N} _{0}}[15] orZ0+{\displaystyle \mathbb {Z} ^{0+}}[16] (for the set including 0).

Intuitive concept

An intuitive and implicit understanding of natural numbers is developednaturally through using numbers for counting, ordering and basic arithmetic.[17] Within this are two closely related aspects of what a natural number is: thesize of a collection;[18] andaposition in a sequence.

Size of a collection

Natural numbers can be used to answer questions like: "how many apples are on the table?".[19] A natural number used in this way describes a characteristic of acollection of objects. This characteristic, thesize of a collection is calledcardinality and a natural number used to describe or measure it is called a cardinal number.

Left: A group of three apples. Right: A group of three oranges.
A group of apples and group of oranges with the same cardinality

Two collections have the same size or cardinality if there is aone-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 thesame cardinality as the group of oranges,[20] 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

Taken together the natural numbers form aninfinite sequence, meaning they have a fixed order, specific starting point and no end point, which is the familiar sequence beginning with 1, 2, 3, and so on indefinitely. A natural number can be used to denote a specific position in any other sequence, in which case it is called anordinal number.[21] 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 oforder.

The natural number 3 then is the thing thatcomes after 2 and 1, andbefore 4, 5 and so on. The number 2 is the thing thatcomes 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.[22]

Counting

Counting is the process of determining the cardinality of a collection of objects by establishing a one-to-one correspondence between the objects to be counted and the sequence of natural numbers starting at 1.[23] 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 (hence they are ordinal numbers) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. The ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the collection.

Formal definitions

Formal definitions take the existing, intuitive notion of natural numbers together with the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. Formal systems typically assume that the defining characteristic of natural numbers is their fixed order[24][25] and establish this order using theprimitive notion of asuccessor. Every natural number has a successor, which is another unique natural number that it is followed by.

Two standard formal definitions are based on thePeano axioms andset theory. The Peano axioms (named forGiuseppe Peano) do not explicitly define what the natural numbersare, but instead comprise a list of statements oraxioms that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particularset, in which a set can be generally understood as a collection of distinct objects orelements. While the two methods are different, they are consistent in that the natural number sets collectivelysatisfy the Peano axioms.

Peano axioms

Main article:Peano axioms

The five Peano axioms are:[26][c]

  1. 0 is a natural number.
  2. Every natural number has a successor which is also a natural number.
  3. 0 is not the successor of any natural number.
  4. If the successor ofx{\displaystyle x} equals the successor ofy{\displaystyle y}, thenx{\displaystyle x} equalsy{\displaystyle y}.
  5. 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 ofx{\displaystyle x} isx+1{\displaystyle x+1}.

Set-theoretic definition

Main articles:Set-theoretic definition of natural numbers andvon Neumann ordinal

In set theory each natural numbern is defined as a specific set. A variety of constructions have been proposed, however the standard solution (due toJohn von Neumann)[27] is:

  • Call0 = { }, theempty set.
  • Define the successorS(a) of any seta byS(a) =a ∪ {a}.
  • By theaxiom of infinity, there exist sets which contain 0 and areclosed under 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.

This produces an iterative definition of the natural numbers called thevon Neumann ordinals:

0={}=∅1=0∪{0}={0}={∅}2=1∪{1}={0,1}={∅,{∅}}3=2∪{2}={0,1,2}={∅,{∅},{∅,{∅}}}n=n−1∪{n−1}={0,1,...,n−1}={∅,{∅},...,{∅,{∅},...}}{\displaystyle {\begin{alignedat}{2}0&&&{}=\{\}&&{}=\varnothing \\1&=0\cup \{0\}&&{}=\{0\}&&=\{\varnothing \}\\2&=1\cup \{1\}&&{}=\{0,1\}&&{}=\{\varnothing ,\{\varnothing \}\}\\3&=2\cup \{2\}&&{}=\{0,1,2\}&&{}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}\\n&=n-1\cup \{n-1\}&&{}=\{0,1,...,n-1\}&&{}=\{\varnothing ,\{\varnothing \},...,\{\varnothing ,\{\varnothing \},...\}\}\\\end{alignedat}}}

In this construction every natural numbern is a set containingn elements, where each element is a natural number less thann. From this, the intuitive concepts of cardinality and order can be formally defined as:

  • Cardinality: a setS hasn elements if there is a one-to-one correspondence orbijection fromn toS.
  • Order:n ≤m if and only ifn is asubset ofm.

Another construction sometimes calledZermelo ordinals[28] defines0 = { } andS(a) = {a} and is now largely only of historical interest.

Properties

This section uses the convention that 0 is a natural number:N=N0{\displaystyle \mathbb {N} =\mathbb {N} _{0}}.

Addition

Given the setN{\displaystyle \mathbb {N} } of natural numbers and thesuccessor functionS:N→N{\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to the next one,addition (+{\displaystyle +}) is defined by:

a+0=a(1)a+S(b)=S(a+b)(2){\displaystyle {\begin{aligned}a+0&=a&{\textrm {(1)}}\\a+S(b)&=S(a+b)&{\textrm {(2)}}\\\end{aligned}}}

In the statements above, (1) explicitly defines addition for the first natural number and (2) gives arecursive definition for each subsequent number in terms of previous definitions, as illustrated below.

a+1=a+S(0)=S(a+0)=S(a)a+2=a+S(1)=S(a+1)=S(S(a))a+3=a+S(2)=S(a+2)=S(S(S(a))){\displaystyle {\begin{alignedat}{2}&a+1=a+S(0)=S(a+0)=S(a)\\&a+2=a+S(1)=S(a+1)=S(S(a))\\&a+3=a+S(2)=S(a+2)=S(S(S(a)))\end{alignedat}}}

In this way, addition can be seen as repeated application of the successor function. Intuitively,a +b is evaluated by applying the successor function toa as many times as it must be applied to0 to produceb.

Thealgebraic structure(N,+){\displaystyle (\mathbb {N} ,+)} is acommutativemonoid withidentity element 0. It is afree monoid on one generator. This commutative monoid satisfies thecancellation property, so it can be embedded in agroup. The smallest group containing the natural numbers is theintegers.

Multiplication

Analogously, given that addition has been defined, amultiplication operator×{\displaystyle \times } can be defined viaa × 0 = 0 anda × S(b) = (a ×b) +a. This turns(N∗,×){\displaystyle (\mathbb {N} ^{*},\times )} into afree commutative monoid with 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 × (b +c) = (a ×b) + (a ×c). These properties of addition and multiplication make the natural numbers an instance of acommutativesemiring. 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 thatN{\displaystyle \mathbb {N} } is notclosed under subtraction (that is, subtracting one natural from another does not always result in another natural), means thatN{\displaystyle \mathbb {N} } isnot aring; 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 witha + 1 =S(a) anda × 1 =a. Furthermore,(N∗,+){\displaystyle (\mathbb {N^{*}} ,+)} has no identity element.

Order

Atotal order on the natural numbers is defined by lettinga ≤b if and only if there exists another natural numberc wherea +c =b. This order is compatible with thearithmetical operations in the following sense: ifa,b andc are natural numbers anda ≤b, thena +c ≤b +c anda ×c ≤b ×c.

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 asω (omega).

Division

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 remainder orEuclidean division is available as a substitute: for any two natural numbersa andb withb ≠ 0 there are natural numbersq andr such that

a=b×q+r and r<b.{\displaystyle a=b\times q+r{\text{ and }}r<b.}

The numberq is called thequotient andr is called theremainder of the division ofa by b. The numbersq andr are uniquely determined bya 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:

  • Closure under addition and multiplication: for all natural numbersa andb, botha +b anda ×b are natural numbers.[29]
  • Associativity: for all natural numbersa,b, andc,a + (b +c) = (a +b) +c anda × (b ×c) = (a ×b) ×c.[30]
  • Commutativity: for all natural numbersa andb,a +b =b +a anda ×b =b ×a.[31]
  • Existence ofidentity elements: for every natural numbera,a + 0 =a anda × 1 =a.
    • If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural numbera,a × 1 =a. However, the "existence of additive identity element" property is not satisfied
  • Distributivity of multiplication over addition for all natural numbersa,b, andc,a × (b +c) = (a ×b) + (a ×c).
  • No nonzerozero divisors: ifa andb are natural numbers such thata ×b = 0, thena = 0 orb = 0 (or both).

History

For most of history, what are now called natural numbers were simplynumbers. Between the late middle ages and end of the 17th century, the concept of number expanded to include negative, rational and irrational numbers, becoming what we now call the real numbers.[32] With this came the need to distinguish between the original numbers and these new types.[33]

Nicolas Chuquet used the termprogression naturelle (natural progression) in 1484.[34] The earliest known use of "natural number" as a complete English phrase is in 1763.[35][36] The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.[36]

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.[37]Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".[d]

Theconstructivists saw a need to improve upon the logical rigor in thefoundations of mathematics.[e] In the 1860s,Hermann Grassmann suggested arecursive 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 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.[40]

In 1881,Charles Sanders Peirce provided the firstaxiomatization of natural-number arithmetic.[41][42] In 1888,Richard Dedekind proposed another axiomatization of natural-number arithmetic,[43] 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 anaxiomatization of the properties ofordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic isequiconsistent with several weak systems ofset theory. One such system isZFC with theaxiom of infinity replaced by its negation.[44] Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms includeGoodstein's theorem.[45]

Zero as natural number

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.[46] In 1889,Giuseppe Peano used N for the positive integers and started at 1,[47] but he later changed to using N0 and N1.[48] Historically, most definitions have excluded 0,[36][49][50] but many mathematicians such asGeorge A. Wentworth,Bertrand Russell,Nicolas Bourbaki,Paul Halmos,Stephen Cole Kleene, andJohn Horton Conway have preferred to include 0.[51][36] This approach gained wider adoption in the 1960s[36] and was formalized inISO 31-11 (1978), which defines natural numbers to include zero, a convention retained in the currentISO 80000-2 standard.[52]

Generalizations

ℕ ⊊ ℤ ⊊ ℚ ⊊ ℝ ⊊ ℂ
Set inclusions between thenatural numbers (ℕ), theintegers (ℤ), therational numbers (ℚ), thereal numbers (ℝ), and thecomplex numbers (ℂ)

The most commonnumber systems used throughout mathematics are extensions of the natural numbers, in the sense that each of them contains a subset which has the same arithmetical structure. These number systems can also be formally defined in terms of natural numbers (though they need not be[f]). If the difference of every two natural numbers is considered to be a number, the result is theintegers, which include zero and negative numbers. If the quotient of every two integers is considered to be a number, the result is therational numbers, includingfractions. If every infinitedecimal is considered to be a number, the result is thereal numbers. If everysolution of a polynomial equation is considered to be a number, the result is thecomplex numbers.

Other generalizations of natural numbers are discussed inNumber § Extensions of the concept.

See also

Notes

  1. ^It depends on authors and context whether 0 is considered a natural number.
  2. ^Older texts have occasionally employedJ as the symbol for this set.[12]
  3. ^Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.  0 is a natural number."
    Halmos (1974, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)  0 ∈ ω (where, of course,0 = ∅" (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)
  4. ^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."[38][39]
  5. ^"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)
  6. ^The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the final construction.[53]

References

  1. ^abEnderton, Herbert B. (1977).Elements of set theory. New York: Academic Press. p. 66.ISBN 0122384407.
  2. ^abCooke, Heather (26 October 2000).Primary Mathematics. SAGE. p. 14.ISBN 978-1-84787-949-3.
  3. ^abZegarelli, Mark (28 January 2014).Basic Math and Pre-Algebra For Dummies. John Wiley & Sons. p. 21.ISBN 978-1-118-79199-8.Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.
  4. ^Woodin, Greg; Winter, Bodo (2024)."Numbers in Context: Cardinals, Ordinals, and Nominals in American English".Cognitive Science.48 (6) e13471.doi:10.1111/cogs.13471.PMC 11475258.PMID 38895756.
  5. ^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."
  6. ^Bluman (2010, p. 1): "Numbers make up the foundation of mathematics."
  7. ^abWeisstein, Eric W."Natural Number".mathworld.wolfram.com. Retrieved11 August 2020.
  8. ^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.Archived from the original on 29 March 2017. Retrieved28 March 2017 – via Google Books.
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  10. ^Weisstein, Eric W."Counting Number".MathWorld.
  11. ^"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions".functions.wolfram.com. Retrieved27 July 2020.
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  14. ^Grimaldi, Ralph P. (2004).Discrete and Combinatorial Mathematics: An applied introduction (5th ed.). Pearson Addison Wesley.ISBN 978-0-201-72634-3.
  15. ^Stewart, Ian; Tall, David (12 March 2015).The Foundations of Mathematics. OUP Oxford. p. 160.ISBN 978-0-19-101648-6. Retrieved30 July 2025.
  16. ^Fokas, Athanassios;Kaxiras, Efthimios (12 December 2022).Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction. World Scientific. p. 4.ISBN 978-1-80061-182-5. Retrieved30 July 2025.
  17. ^Mayberry (2000), p. xvi: "We gain our knowledge of these numbers when we learn to count them out and to calculate with them, so we are led to see these processes of counting out and calculating as constitutive of the very notion of natural number."
  18. ^Tao (2016), p. 68: says, "...one of our main conceptualizations of natural numbers (is) that of cardinality, or measuring how many elements there are in a set".
  19. ^Frege, Gottlob; Frege, Gottlob (1975) [1953].The foundations of arithmetic: a logico-mathematical enquiry into the concept of number (2. revised ed.). Evanston Ill: Northwestern Univ. Press. p. 5.ISBN 978-0-8101-0605-5.
  20. ^Mayberry (2000), p. 135: "Cantor's discovery was that ...'sameness of size', defined in terms of one-to-one correspondence, is logically prior to the notion of counting or, indeed, to the notion of 'number'..."
  21. ^Tao, Terence (2016).Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer Singapore.doi:10.1007/978-981-10-1789-6.ISBN 978-981-10-1789-6.
  22. ^Benacerraf, Paul (January 1965)."What Numbers Could not Be".The Philosophical Review.74 (1):47–73.doi:10.2307/2183530.JSTOR 2183530.
  23. ^Russell (2012), p. 13
  24. ^Tao (2016), p. 68: "Indeed, the Peano axiom approach treats natural numbers more like ordinals than cardinals."
  25. ^Russell, Bertrand (18 December 2012).Introduction to Mathematical Philosophy.
  26. ^Mints, G.E. (ed.)."Peano axioms".Encyclopedia of Mathematics.Springer, in cooperation with theEuropean Mathematical Society.Archived from the original on 13 October 2014. Retrieved8 October 2014.
  27. ^von Neumann (1923)
  28. ^Levy (1979), p. 52
  29. ^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
  30. ^Davisson, Schuyler Colfax (1910).College Algebra. Macmillian Company. p. 2.Addition of natural numbers is associative.
  31. ^Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.
  32. ^Mayberry (2000), p. 17
  33. ^Ifrah, Georges (9 October 2000).The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley.ISBN 978-0-471-39340-5.
  34. ^Chuquet, Nicolas (1881) [1484].Le Triparty en la science des nombres (in French).
  35. ^Emerson, William (1763).The method of increments. p. 113.
  36. ^abcde"Earliest Known Uses of Some of the Words of Mathematics (N)".Maths History.
  37. ^Poincaré, Henri (1905) [1902]."On the nature of mathematical reasoning".La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.
  38. ^Gray, Jeremy (2008).Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153.ISBN 978-1-4008-2904-0.Archived from the original on 29 March 2017 – via Google Books.
  39. ^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 original on 9 August 2018;"access toJahresbericht der Deutschen Mathematiker-Vereinigung". Archived fromthe original on 20 August 2017.
  40. ^Eves 1990, Chapter 15
  41. ^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.
  42. ^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.
  43. ^Was sind und was sollen die Zahlen? (in German). F. Vieweg. 1893. 71–73.
  44. ^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.
  45. ^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.
  46. ^Fontenelle, Bernard de (1727).Eléments de la géométrie de l'infini (in French). p. 3.
  47. ^Arithmetices principia: nova methodo (in Latin). Fratres Bocca. 1889. p. 12.
  48. ^Peano, Giuseppe (1901).Formulaire des mathematiques (in French). Paris, Gauthier-Villars. p. 39.
  49. ^Fine, Henry Burchard (1904).A College Algebra. Ginn. p. 6.
  50. ^Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166. United States Armed Forces Institute. 1958. p. 12.
  51. ^"Natural Number".archive.lib.msu.edu.
  52. ^"Standard number sets and intervals"(PDF).ISO 80000-2:2019 Quantities and units Part 2: Mathematics.International Organization for Standardization. 24 June 2025.
  53. ^Wohlgemuth, Andrew (10 June 2014).Introduction to Proof in Abstract Mathematics. Courier Corporation. p. 237.ISBN 978-0-486-14168-8.

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