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

Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with 1, corresponding to the positive integers .

says: "ℕ is the set of natural numbers (positive integers)" (p. 3) include zero in the natural numbers: 'Intuitively, the set of all natural numbers may be described as follows: contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15) 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).

(2018). 9781578201204 .

The natural numbers are a basis from which many other number sets may be built by extension: the (Grothendieck group), by including (if not yet in) the 0 and an (− n) for each nonzero natural number n; the , by including a multiplicative inverse (1/ n) for each nonzero integer n (and also the product of these inverses by integers); the by including with the rationals the limits of (converging) Cauchy sequences of rationals; the , by including with the real numbers the unresolved (and also the sums and products thereof); and so on. 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): "Numbers make up the foundation of mathematics." (p. 1) These chains of extensions make the natural numbers canonically (identified) in the other number systems.

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

In common language, for example in , natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of to the continuity of , established by the .

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


History

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

The first major advance in abstraction was the use of 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 , dating from around 1500 BC and now at the in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The 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.Georges Ifrah, The Universal History of Numbers, Wiley, 2000,

A much later advance was the development of the idea that  can be considered as a number, with its own numeral. The use of a 0 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. The and Maya civilizations used 0 as a separate number as early as the , but this usage did not spread beyond ... The use of a numeral 0 in modern times originated with the mathematician in 628. However, 0 had been used as a number in the medieval (the calculation of the date of ), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.

The first systematic study of numbers as is usually credited to the philosophers and . Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.

Independent studies also occurred at around the same time in , , and .Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 1972,


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

Weber, Heinrich L. 1891-2. Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 2:5-23. (The quote is on p. 19.)

In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics."Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." 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 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.

The second class of definitions was introduced by Charles Sanders Peirce, refined by , and further explored by ; this approach is now called . It is based on an of the properties of : each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is with several weak systems of set theory. One such system is 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.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.

With all these definitions it is convenient to include 0 (corresponding to the ) as a natural number. Including 0 is now the common convention among and .

(1998). 9780412606106, Chapman & Hall/CRC.
Other mathematicians also include 0, e.g., Computer languages often start from zero when enumerating items like and string- or array-elements. Many mathematicians have kept the older tradition and take 1 to be the first natural number.This is common in texts about . See, for example, or .

Since different properties are customarily associated to the tokens and , e.g., neutral elements for addition and multiplications, respectively, it is important to know which version of natural numbers, generically denoted by \mathbb N, 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),

9780201726343, Pearson Addison Wesley.
e.g., like this:
  • Naturals with zero: \;\{0,1,2,...\}=\mathbb{N}_0={\mathbb{N}}\cup\{0\}
  • Naturals without zero: \{1,2,...\}=\mathbb{N}^*=\mathbb{N}\smallsetminus\{0\}.


Notation
Mathematicians use N or (an N in ) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.
(1976). 9780070542358, McGraw-Hill. .

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 "" is added in the latter case:

.

Alternatively, since natural numbers naturally in the , they may be referred to as the positive, or the non-negative integers, respectively.

(2018). 9780201726343, Addison-Wesley.

\{1, 2, 3,\dots\} = \mathbb Z^+
\{0, 1, 2,\dots\} = \mathbb Z^{\ge 0}


Properties

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


Addition
One can recursively define an addition operator on the natural numbers by setting and for all , . Here should be read as "successor". This turns the natural numbers into a with  0, the so-called 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 .

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


Multiplication
Analogously, given that addition has been defined, a operator × can be defined via and . This turns into a free commutative monoid with identity element 1; a generator set for this monoid is the set of .


Relationship between addition and multiplication
Addition and multiplication are compatible, which is expressed in the : . These properties of addition and multiplication make the natural numbers an instance of a . Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that is not a ring; instead it is a (also known as a rig).

If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with and .


Order
In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed.

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

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


Division
In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers and with there are natural numbers and such that

     and      .

The number is called the and is called the of the division of by . The numbers and are uniquely determined by and . This Euclidean division is key to several other properties (), 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 and , both and are natural numbers.
  • : for all natural numbers , , and , and .
  • : for all natural numbers and , and .
  • Existence of : for every natural number a, and .
  • of multiplication over addition for all natural numbers , , and , .
  • No nonzero : if and are natural numbers such that , then or (or both).


Generalizations
Two important generalizations of natural numbers arise from the two uses of counting and ordering: and .
  • 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 , exactly if there exists a between them. The set of natural numbers itself, and any bijective image of it, is said to be and to have aleph-null ().
  • 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 countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ; this is also the ordinal number of the set of natural numbers itself.

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

For 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, .

A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by in 1933. The numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.

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


Formal definitions

Peano axioms
Many properties of the natural numbers can be derived from the five : calls them "Peano's Postulates" and begins with "1.0 is a natural number." (p. 117f)
uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) (where, of course, " ( is the set of all natural numbers). (p. 46)
gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')

  1. 0 is a natural number.
  2. Every natural number has a successor.
  3. 0 is not the successor of any natural number.
  4. If the successor of x equals the successor of y , then x equals y.
  5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of x is x + 1. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.


Constructions based on set theory

Von Neumann ordinals
In the area of mathematics called , a specific construction due to John von Neumann attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s. defines the natural numbers as follows:
  • Set , the ,
  • Define for every set . is the successor of , and is called the successor function.
  • By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the .
  • It follows that each natural number is equal to the set of all natural numbers less than it:
*,
*,
*,
*,
*, etc.

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

Also, with this definition, different possible interpretations of notations like (-tuples versus mappings of into ) coincide.

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


Zermelo ordinals
Although the standard construction is useful, it is not the only possible construction. 's construction goes as follows:
  • Set
  • Define ,
  • It then follows that
*,
*,
*,
*, etc.
Each natural number is then equal to the set containing just the natural number preceding it. (This is the definition of Zermelo ordinals.)


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


Notes
    • - English translation of .


External links

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