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# Natural number

In mathematics, the **natural numbers** are those used for counting (as in "there are *six* coins on the table") and ordering (as in "this is the *third* largest city in the country"). In common mathematical terminology, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

Some definitions, including the standard ISO 80000-2,^{[1]} begin the natural numbers with 0, corresponding to the **non-negative integers** 0, 1, 2, 3, …, whereas others start with 1, corresponding to the **positive integers** 1, 2, 3, ….^{[2]}^{[3]}^{[4]}^{[5]} Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the **whole numbers**, but in other writings, that term is used instead for the integers (including negative integers).^{[6]}

The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−*n*) for each nonzero natural number *n*; the rational numbers, by including a multiplicative inverse (1/*n*) for each nonzero integer *n* (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on.^{[7]}^{[8]} These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

In common language, for example in primary school, natural numbers may be called **counting numbers**^{[9]} both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.

The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.

## Contents

## History[edit | edit source]

### Ancient roots[edit | edit source]

The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.

The first major advance in abstraction was the use of numerals 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 the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC 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.^{[13]}

A much later advance was the development of the idea that 0 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 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.^{[14]} The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.^{[15]}^{[16]} The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. 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, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead *nulla* (or the genitive form *nullae*) from *nullus*, the Latin word for "none", was employed to denote a 0 value.^{[17]}

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.^{[18]}

Independent studies also occurred at around the same time in India, China, and Mesoamerica.^{[19]}

### Modern definitions[edit | edit source]

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school^{[which?]} of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".^{[20]}

In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.^{[21]} 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 were constructed; later, they were shown to be equivalent in most practical applications.

Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this 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.^{[22]}

The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.^{[23]}

With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists^{[24]} and logicians.^{[25]} Other mathematicians also include 0^{[5]}, e.g., Computer languages often start from zero when enumerating items like loop counters and string- or array-elements.^{[26]}^{[27]} Many mathematicians have kept the older tradition and take 1 to be the first natural number.^{[28]}

Since different properties are customarily associated to the tokens 0 and 1, e.g., neutral elements for addition and multiplications, respectively, it is important to know which version of *natural numbers*, generically denoted by '"`UNIQ--postMath-00000001-QINU`"' is employed in the case under consideration. This can be done by explanation in prose, by explicitly writing down the set, or by qualifying the generic identifier with a super- or subscript (see also in #Notation),^{[1]}^{[29]} e.g., like this:

- Naturals with zero: '"`UNIQ--postMath-00000002-QINU`"'
- Naturals without zero: '"`UNIQ--postMath-00000003-QINU`"'

## Notation[edit | edit source]

Mathematicians use **N** or ℕ (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed *J* as the symbol for this set.^{[30]}

To be unambiguous about whether 0 is included or not, sometimes a subscript (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:^{[1]}

- ℕ
^{0}= ℕ_{0}= {0, 1, 2, …} - ℕ
^{*}= ℕ^{+}= ℕ_{1}= ℕ_{>0}= {1, 2, 3, …}.

Alternatively, since natural numbers naturally embed in the integers, they may be referred to as the positive, or the non-negative integers, respectively.^{[31]}

- '"`UNIQ--postMath-00000004-QINU`"'
- '"`UNIQ--postMath-00000005-QINU`"'

## Properties[edit | edit source]

### Infinity[edit | edit source]

The set of natural numbers is an infinite set. This kind of infinity is, by definition, called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-naught (ℵ_{0}).^{[32]}

### Addition[edit | edit source]

One can recursively define an addition operator on the natural numbers by setting *a* + 0 = *a* and *a* + *S*(*b*) = *S*(*a* + *b*) for all *a*, *b*. Here *S* should be read as "successor". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word *group*). The smallest group containing the natural numbers is the integers.

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[edit | edit source]

Analogously, given that addition has been defined, a multiplication operator × can be defined via *a* × 0 = 0 and *a* × S(*b*) = (*a* × *b*) + *a*. This turns (ℕ^{*}, ×) 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[edit | edit source]

Addition and multiplication are compatible, which is expressed in the distribution law: *a* × (*b* + *c*) = (*a* × *b*) + (*a* × *c*). 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 (i.e., 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 *a* + 1 = *S*(*a*) and *a* × 1 = *a*.

### Order[edit | edit source]

In this section, juxtaposed variables such as *ab* indicate the product *a* × *b*, and the standard order of operations is assumed.

A total order on the natural numbers is defined by letting *a* ≤ *b* if and only if there exists another natural number *c* where *a* + *c* = *b*. This order is compatible with the arithmetical operations in the following sense: if *a*, *b* and *c* are natural numbers and *a* ≤ *b*, then *a* + *c* ≤ *b* + *c* and *ac* ≤ *bc*.

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 *ω* (omega).

### Division[edit | edit source]

In this section, juxtaposed variables such as *ab* indicate the product *a* × *b*, 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* is available as a substitute: for any two natural numbers *a* and *b* with *b* ≠ 0 there are natural numbers *q* and *r* such that

*a*=*bq*+*r*and*r*<*b*.

The number *q* is called the *quotient* and *r* is called the *remainder* of the division of *a* by *b*. The numbers *q* and *r* are uniquely determined by *a* and *b*. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

### Algebraic properties satisfied by the natural numbers[edit | edit source]

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

- Closure under addition and multiplication: for all natural numbers
*a*and*b*, both*a*+*b*and*a*×*b*are natural numbers. - Associativity: for all natural numbers
*a*,*b*, and*c*,*a*+ (*b*+*c*) = (*a*+*b*) +*c*and*a*× (*b*×*c*) = (*a*×*b*) ×*c*. - Commutativity: for all natural numbers
*a*and*b*,*a*+*b*=*b*+*a*and*a*×*b*=*b*×*a*. - Existence of identity elements: for every natural number
*a*,*a*+ 0 =*a*and*a*× 1 =*a*. - Distributivity of multiplication over addition for all natural numbers
*a*,*b*, and*c*,*a*× (*b*+*c*) = (*a*×*b*) + (*a*×*c*). - No nonzero zero divisors: if
*a*and*b*are natural numbers such that*a*×*b*= 0, then*a*= 0 or*b*= 0 (or both).

## Generalizations[edit | edit source]

Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.

- A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have 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
*countably infinite*and to have cardinality aleph-null (ℵ_{0}). - Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.

Many well-ordered sets with cardinal number ℵ_{0} have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ_{0} (i.e., the initial ordinal) is ω.

For 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 (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.

Georges Reeb used to claim provocatively that *The naïve integers don't fill up* ℕ. Other generalizations are discussed in the article on numbers.

## Formal definitions[edit | edit source]

### Peano axioms[edit | edit source]

Many properties of the natural numbers can be derived from the five Peano axioms:^{[33]}^{[34]}

- 0 is a natural number.
- Every natural number has a successor.
- 0 is not the successor of any natural number.
- If the successor of '"`UNIQ--postMath-00000006-QINU`"' equals the successor of '"`UNIQ--postMath-00000007-QINU`"', then '"`UNIQ--postMath-00000008-QINU`"' equals '"`UNIQ--postMath-00000009-QINU`"'.
- 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 '"`UNIQ--postMath-0000000A-QINU`"' is '"`UNIQ--postMath-0000000B-QINU`"'. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called *Peano Arithmetic.*

### Constructions based on set theory[edit | edit source]

#### Von Neumann ordinals[edit | edit source]

In the area of mathematics called set theory, a specific construction due to John von Neumann^{[35]}^{[36]} defines the natural numbers as follows:

- Set 0 = { }, the empty set,
- Define
*S*(*a*) =*a*∪ {*a*} for every set*a*.*S*(*a*) is the successor of*a*, and*S*is called the successor function. - By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.
- It follows that each natural number is equal to the set of all natural numbers less than it:

- 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} = {{ }, {{ }}, …, {{ }, {{ }}, …}}, etc.

With this definition, a natural number *n* is a particular set with *n* elements, and *n* ≤ *m* if and only if *n* is a subset of *m*. The standard definition, now called definition of **von Neumann ordinals**, is: "each ordinal is the well-ordered set of all smaller ordinals."

Also, with this definition, different possible interpretations of notations like ℝ^{n} (*n*-tuples versus mappings of *n* into ℝ) coincide.

Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

#### Zermelo ordinals[edit | edit source]

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction^{[36]} goes as follows:

- Set 0 = { }
- Define
*S*(*a*) = {*a*}, - It then follows that

- 0 = { },
- 1 = {0} = {{ }},
- 2 = {1} = {{{ }}},
*n*= {*n*−1} = {{{…}}}, etc.

- Each natural number is then equal to the set containing just the natural number preceding it. (This is the definition of
**Zermelo ordinals**.)

## See also[edit | edit source]

Others articles of the Topic **Mathematics** : List of positive integers and factors/4, The Number Decomposing Formula or GOF(Getal Ontledingsformule), Applied mathematics, James Grime, List of positive integers and factors 2001 to 4000, List of positive integers and factors 4001 to 6000, Integer

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

## Notes[edit | edit source]

- ↑
^{1.0}^{1.1}^{1.2}"Standard number sets and intervals".*ISO 80000-2:2009*. International Organization for Standardization. p. 6. - ↑
Weisstein, Eric W. "Natural Number".
*MathWorld*. - ↑
"natural number",
*Merriam-Webster.com*, Merriam-Webster, retrieved 4 October 2014 - ↑ Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)
- ↑
^{5.0}^{5.1}Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all*natural numbers*may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15) - ↑ Jack G. Ganssle & Michael Barr (2003).
*Embedded Systems Dictionary*. p. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 1578201209.integer 1. n. Any whole number.

- ↑ Mendelson (2008) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)
- ↑ Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)
- ↑ Weisstein, Eric W. "Counting Number".
*MathWorld*. - ↑ "Introduction".
*Royal Belgian Institute of Natural Sciences*. Brussels, Belgium. Archived from the original on March 4, 2016. - ↑ Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
- ↑ The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of Astronomy
- ↑ Georges Ifrah,
*The Universal History of Numbers*, Wiley, 2000, ISBN 0-471-37568-3 - ↑ "A history of Zero".
*MacTutor History of Mathematics*. Retrieved 2013-01-23.… 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

- ↑ Mann, Charles C. (2005),
*1491: New Revelations Of The Americas Before Columbus*, Knopf, p. 19, ISBN 9781400040063. - ↑ Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations",
*The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context*, John Wiley & Sons, ISBN 9781118853979. - ↑ Michael Deckers (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Retrieved 2012-02-13.
- ↑ This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.
- ↑ Morris Kline,
*Mathematical Thought From Ancient to Modern Times*, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7 - ↑ The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."

Gray, Jeremy (2008),*Plato's Ghost: The Modernist Transformation of Mathematics*, Princeton University Press, p. 153

Weber, Heinrich L. 1891-2. Kronecker.*Jahresbericht der Deutschen Mathematiker-Vereinigung*2:5-23. (The quote is on p. 19.) - ↑ "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)
- ↑ Eves 1990, Chapter 15
- ↑ L. Kirby; J. Paris,
*Accessible Independence Results for Peano Arithmetic*, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982. - ↑ Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).
- ↑ 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 0-412-60610-0. - ↑ Brown, Jim (1978). "In Defense of Index Origin 0".
*ACM SIGAPL APL Quote Quad*.**9**(2): 7. doi:10.1145/586050.586053. Retrieved 19 January 2015. - ↑ Hui, Roger. "Is Index Origin 0 a Hindrance?".
*jsoftware.com*. Retrieved 19 January 2015. - ↑ This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).
- ↑ Grimaldi, Ralph P.
*Discrete and combinatorial mathematics : an applied introduction*(5th ed.). Pearson Addison Wesley. ISBN 0201726343. - ↑ Rudin, W. (1976).
*Principles of Mathematical Analysis*. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8. - ↑ Grimaldi, Ralph P. (2003).
*A review of discrete and combinatorial mathematics*(5th ed.). Boston, MA: Addison-Wesley. p. 133. ISBN 978-0201726343. - ↑ Weisstein, Eric W. "Cardinal Number".
*MathWorld*. - ↑ G.E. Mints (originator), "Peano axioms",
*Encyclopedia of Mathematics*, Springer, in cooperation with the European Mathematical Society, retrieved 8 October 2014 - ↑ Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f)

Halmos (1960) 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). (p. 46)

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*) - ↑ Von Neumann 1923
- ↑
^{36.0}^{36.1}(Levy 1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.

## References[edit | edit source]

- Bluman, Allan (2010),
*Pre-Algebra DeMYSTiFieD*(Second ed.), McGraw-Hill Professional - Carothers, N.L. (2000),
*Real analysis*, Cambridge University Press, ISBN 0-521-49756-6 - Clapham, Christopher; Nicholson, James (2014),
*The Concise Oxford Dictionary of Mathematics*(Fifth ed.), Oxford University Press - Dedekind, Richard (1963),
*Essays on the Theory of Numbers*, Dover, ISBN 0-486-21010-3- Dedekind, Richard (2007),
*Essays on the Theory of Numbers*, Kessinger Publishing, LLC, ISBN 0-548-08985-X

- Dedekind, Richard (2007),
- Eves, Howard (1990),
*An Introduction to the History of Mathematics*(6th ed.), Thomson, ISBN 978-0-03-029558-4 - Halmos, Paul (1960),
*Naive Set Theory*, Springer Science & Business Media - Hamilton, A. G. (1988),
*Logic for Mathematicians*(Revised ed.), Cambridge University Press - James, Robert C.; James, Glenn (1992),
*Mathematics Dictionary*(Fifth ed.), Chapman & Hall - Landau, Edmund (1966),
*Foundations of Analysis*(Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-X - Mac Lane, Saunders; Birkhoff, Garrett (1999),
*Algebra*(3rd ed.), American Mathematical Society - Mendelson, Elliott (2008) [1973],
*Number Systems and the Foundations of Analysis*, Dover Publications - Morash, Ronald P. (1991),
*Bridge to Abstract Mathematics: Mathematical Proof and Structures*(Second ed.), Mcgraw-Hill College - Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013),
*Mathematics for Elementary Teachers: A Contemporary Approach*(10th ed.), Wiley Global Education, ISBN 978-1118457443 - Szczepanski, Amy F.; Kositsky, Andrew P. (2008),
*The Complete Idiot's Guide to Pre-algebra*, Penguin Group - Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008),
*Elementary Real Analysis*(Second ed.), ClassicalRealAnalysis.com, ISBN 9781434843678 - Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen",
*Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum*,**1**: 199–208- Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort,
*From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931*(3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.

- Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort,

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