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| | For quite a while now, individuals have advised, don't believe everything you read. Well, together with the beginning of the web, this relatively cynical caution couldn't be more sound guidance. The net has, essentially, become one big discussion board. But, amidst the ideas of 12-year-olds on Woman Coo's newest fashion alternative, there is some dependable literature out there, right?<br><br>So, how will you get started on this? Where would you begin? There's a lot to learn, no doubt about any of it. However, if you take the time to master [https://www.larrainvial.com/Content/descargas/fondos/FI/prospecto_deuda_subsidio.pdf jaun pablo schiappacasse canepa], it will set you up for a lifetime.<br><br>When someone buys something from your link then you definitely'll get an area of the value. If the affiliate program gives out a 50% payment and the item sells for $100.00 your slice will be $50.00 The stunning thing about affiliate marketing is the fact that it is all done online consequently you don't have to do any one-on-one selling or often no selling at all. All you've got to complete, generally, is just expose interested visitors to the merchandise. And everything happens online. As you may promote on the Internet frequently at no cost or very reasonably, it's an ideal place to start. You'll have the ability to start slowly, in your spare-time and as you learn how to earn online your organization may increase to other online projects. It might develop into a supply of passive income.<br><br><br><br>That is another quite a familiar sort of internet marketing. When you search well for a specific website, you are sometimes warned by your browser about this having blocked a pop up. At different moments, the moment you enter an internet site, before you can view this content you wish to, you are designed to view a pop up advertising of a particular organization. This sort of web marketing is known as intrusive and consequently surfers are allowed with blockers.<br><br>Located in a filthy fraction next-to the Old City's Blue Mosque, Ka Firushi Hen industry is indeed accustomed to bird-owners. In small open-fronted shops, other, bulbuls, budgerigars and canaries songbirds are sold by shopkeepers.<br><br>Advertising is about making the consumer feel truly special, since no one likes to be one of the crowd. Generally customize your communication to help make the clients feel more specific. Handle it right to a particular person as an example, To: Mr. John Smith. In the event you really want to cheer the consumers, send along a free gift to seize their interest and produce a 'feel - good' factor.<br><br>Here are a few handy suggestions to enable you to get started on the path to greater presence online for your goods. Many involve only the expense of time, and won't destroy your financial allowance.<br><br>Both List Joe and Viral URL give you a large amount of of freedom to use of their respective networks. Both have invested significant profit making integrated, effective marketing with email application which you can not get as an individual home based business. In our experience, we recommend getting both plans, as they offer you use of a larger world of customers that are previously decided in and applied to obtaining these kinds of offers. |
| {{Refimprove|date=October 2011}}
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| {{more footnotes|date=December 2013}}
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| }}
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| [[File:Three apples.svg|right|thumb|Natural numbers can be used for counting (one [[apple]], two apples, three apples, ...)]]
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| In [[mathematics]], the '''natural numbers''' are those used for [[counting]] ("there are six coins on the table") and [[total order|ordering]] ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see [[English numerals]]). A later notion is that of a [[nominal number]], which is used only for naming.
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| Properties of the natural numbers related to [[divisibility]], such as the distribution of [[prime number]]s, are studied in [[number theory]]. Problems concerning counting and ordering, such as [[Partition (number theory)|partition]] [[enumeration]], are studied in [[combinatorics]].
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| There is no universal agreement about whether to include [[zero]] in the set of natural numbers: some define the natural numbers to be the '''[[positive number|positive]] [[integer]]s''' {{nowrap|1={{{num|1}}, {{num|2}}, {{num|3}}, ...}}}, while for others the term designates the '''[[non-negative]] integers''' {{nowrap|1={{{num|0}}, 1, 2, 3, ...}}}. The former definition is the traditional one, with the latter definition having first appeared in the 19th century.
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| ==History of natural numbers and the status of zero==
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| <!--This section is linked from [[Numeral system]]-->
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| The most primitive method of representing a natural number is to put down a dot for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a dot for each object in the set.
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| The first major advance in abstraction was the use of [[numeral system|numerals]] to represent numbers. This allowed systems to be developed for recording large numbers. The ancient [[History of Ancient Egypt|Egyptians]] developed a powerful system of numerals with distinct [[Egyptian hieroglyphs|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 [[Babylonia]]ns had a [[Positional notation|place-value]] system based essentially on the numerals for 1 and 10.{{citation needed|date=November 2012}} | |
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| A much later advance was the development of the idea that {{num|0}} can be considered as a number, with its own numeral. The use of a 0 [[numerical digit|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.<ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |title=A history of Zero |quote=... 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 |website=MacTutor History of Mathematics |accessdate=2013-01-23}}</ref> The [[Olmec]] and [[Maya civilization]]s used 0 as a separate number as early as the {{nowrap|1st century BC}}, but this usage did not spread beyond [[Mesoamerica]].{{Citation needed|date=October 2012}} The use of a numeral 0 in modern times originated with the [[India]]n 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 ''nullae'', genitive of ''nullus'', the Latin word for "none", was employed to denote a 0 value.<ref>{{cite web |author=Michael L. Gorodetsky |url=http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |title=Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius |publisher=Hbar.phys.msu.ru |date=2003-08-25 |accessdate=2012-02-13}}</ref>
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| The first systematic study of numbers as [[abstraction]]s (that is, as abstract [[entity|entities]]) is usually credited to the [[ancient Greece|Greek]] philosophers [[Pythagoras]] and [[Archimedes]]. Many Greek mathematicians did not consider 1 to be "a number", so to them '''2''' was the smallest number.<ref>This convention is used, for example, in [[Euclid's Elements]], see [http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html Book VII, definitions 1 and 2].</ref>
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| Independent studies also occurred at around the same time in [[India]], [[China]], and [[Mesoamerica]].{{Citation needed|date=October 2011}}
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| Several [[set-theoretical definitions of natural numbers]] were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the [[empty set]]) as a natural number. Including 0 is now the common convention among [[set theory|set theorists]], [[logic]]ians, and [[computer science|computer scientists]]. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number.<ref>This is common in texts about [[Real analysis]]. See, for example, Carothers (2000) p.3 or Thomson, Bruckner and Bruckner (2000), p.2.</ref> The term ''counting number'' is also used to refer to the natural numbers (either including or excluding 0). Likewise, some authors use the term ''whole number'' to mean a natural number including 0; some use it to mean a natural number excluding 0; while others use it in a way that includes both 0 and the negative integers, as an equivalent of the term ''integer''.<ref>{{MathWorld|title=Counting Number|id=CountingNumber|title2=Whole Number|id2=WholeNumber}}</ref>
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| In using [[Indexed family|indices]], beginning with 1 is still common (there is no 0-th row or column of a matrix), except when (as is often the case) the beginning is a rather trivial form of the problem fittingly numbered zero (the first and second [[Fibonacci number]] are 1 whatever form of <math>\mathbb{N}</math> is used, yet it is convenient, and often done, to begin with a 0th Fibonacci number, which is 0). In all areas of mathematics that deal with [[space (mathematics)|spaces]] of some sort, <math>\mathbb{N}</math> as a rule includes zero, because (e.g.) <math>\mathbb{R}^0</math> is the logical (even if in itself trivial) beginning of a series of spaces of ascending dimension.
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| ==Notation==
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| Mathematicians use '''N''' or <math>\mathbb{N}</math> (an N in [[blackboard bold]], displayed as {{unicode|ℕ}} in [[Unicode]]) to refer to the [[Set (mathematics)|set]] of all natural numbers. This set is countably infinite: it is [[infinite set|infinite]] but [[countable set|countable]] by definition. This is also expressed by saying that the [[cardinal number]] of the set is [[Aleph number#Aleph-null|aleph-null]] <math>(\aleph_0)</math>.<ref>{{MathWorld |urlname=CardinalNumber |title=Cardinal Number}}</ref>
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| To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "<math>*</math>" or subscript "<math>1</math>" is added in the latter case:{{citation needed|date=May 2013}}
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| :<math>\mathbb{N}^0 = \mathbb{N}_0 = \{ 0, 1, 2, \ldots \}</math>
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| :<math>\mathbb{N}^* = \mathbb{N}^+ = \mathbb{N}_1 = \mathbb{N}_{>0}= \{ 1, 2, \ldots \}. </math>
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| <!--(Sometimes, an index or [[superscript]] "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as '''R'''<sup>+</sup> = <nowiki>[0,∞)</nowiki> and '''Z'''<sup>+</sup> = { 0, 1, 2, ... }, but rarely in European scientific journals. The notation "<math>*</math>", however, is standard for nonzero, or rather, [[invertible]] elements. The notation <math>\mathbb{N}^0</math> could also mean the empty [[direct product]] <math>\prod_{i=1}^k \mathbb{N}</math> resp. the empty [[direct sum]] <math>\bigoplus_{i=1}^k \mathbb{N}</math> in the case <math>k=0</math>.)-->
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| Some authors who exclude 0 from the naturals may distinguish the set of nonnegative integers by referring to the latter as the ''natural numbers with zero'', ''whole numbers'', or ''counting numbers'', denoted '''W'''.{{citation needed|date=May 2013}} Others use the notation '''P''' for the positive integers if there is no danger of confusing this with the prime numbers.{{citation needed|date=May 2013}} In that case, a popular{{citation needed|date=May 2013}} notation is to use a script '''''P''''' for positive integers (which extends to using script '''''N''''' for negative integers, and script '''''Z''''' for 0).
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| Set theorists often denote the set of all natural numbers including 0 by a lower-case Greek letter [[omega]]: ''ω''. This stems from the identification of an [[ordinal number]] with the set of ordinals that are smaller. Moreover, adopting the [[von Neumann definition of ordinals]] and defining cardinal numbers as minimal ordinals among those with same [[cardinality]] leads to the identity <math>\,\mathbb N_0=\aleph_0=\omega</math>.
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| ==Algebraic properties==
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| The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:
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| *[[Closure (mathematics)|Closure]] under addition and multiplication: for all natural numbers ''a'' and ''b'', both {{nowrap|''a'' + ''b''}} and {{nowrap|''a'' × ''b''}} are natural numbers.
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| *[[Associativity]]: for all natural numbers ''a'', ''b'', and ''c'', {{nowrap|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}} and {{nowrap|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}.
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| *[[Commutativity]]: for all natural numbers ''a'' and ''b'', {{nowrap|''a'' + ''b'' {{=}} ''b'' + ''a''}} and {{nowrap|''a'' × ''b'' {{=}} ''b'' × ''a''}}.
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| *Existence of [[identity element]]s: for every natural number ''a'', {{nowrap|''a'' + 0 {{=}} ''a''}} and {{nowrap|''a'' × 1 {{=}} ''a''}}.
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| *[[Distributivity]] of multiplication over addition for all natural numbers ''a'', ''b'', and ''c'', {{nowrap|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}.
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| *No nonzero [[zero divisor]]s: if ''a'' and ''b'' are natural numbers such that {{nowrap|''a'' × ''b'' {{=}} 0}}, then {{nowrap|''a'' {{=}} 0}} or {{nowrap|''b'' {{=}} 0}}.
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| ==Properties==
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| One can recursively define an [[Addition in N|addition]] on the natural numbers by setting {{nowrap|''a'' + 0 {{=}} ''a''}} and {{nowrap|''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 (mathematics)|group]] (in the mathematical sense of the word ''group''). The smallest group containing the natural numbers is the [[integer]]s.
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| If 1 is defined as ''S''(0), then {{nowrap|''b'' + 1 {{=}} ''b'' + ''S''(0) {{=}} ''S''(''b'' + 0) {{=}} ''S''(''b'')}}. That is, {{nowrap|''b'' + 1}} is simply the successor of ''b''.
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| Analogously, given that addition has been defined, a [[multiplication]] × can be defined via {{nowrap|''a'' × 0 {{=}} 0}} and {{nowrap|''a'' × S(''b'') {{=}} (''a'' × ''b'') + ''a''}}. This turns ('''N'''<sup>*</sup>, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of [[prime number]]s. Addition and multiplication are compatible, which is expressed in the [[distributivity|distribution law]]: {{nowrap|''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, means that '''N''' is ''not'' a [[ring (mathematics)|ring]]; instead it is a [[semiring]] (also known as a ''rig'').
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| 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 {{nowrap|''a'' + 1 {{=}} ''S''(''a'')}} and {{nowrap|''a'' × 1 {{=}} ''a''}}.
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| For the remainder of the article, juxtaposed variables such as ''ab'' indicate the product ''a'' × ''b'', and the standard [[order of operations]] is assumed.
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| A [[total order]] on the natural numbers is defined by letting {{nowrap|''a'' ≤ ''b''}} if and only if there exists another natural number ''c'' with {{nowrap|''a'' + ''c'' {{=}} ''b''}}. This order is compatible with the [[arithmetical operations]] in the following sense: if ''a'', ''b'' and ''c'' are natural numbers and {{nowrap|''a'' ≤ ''b''}}, then {{nowrap|''a'' + ''c'' ≤ ''b'' + ''c''}} and {{nowrap|''ac'' ≤ ''bc''}}. An important property of the natural numbers is that they are [[well-order]]ed: 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 expressed as ''ω''.
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| 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 (mathematics)|division]] with remainder'' is available as a substitute: for any two natural numbers ''a'' and ''b'' with {{nowrap|''b'' ≠ 0}} there are natural numbers ''q'' and ''r'' such that
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| :''a'' = ''bq'' + ''r'' and ''r'' < ''b''.
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| The number ''q'' is called the ''[[quotient]]'' and ''r'' is called the ''[[remainder]]'' of 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.
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| ==Generalizations==
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| Two generalizations of natural numbers arise from the two uses:
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| *A natural number can be used to express the size of a finite set; more generally a [[cardinal number]] is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have [[equinumerosity|the same size]]. The set of natural numbers itself and any other countably infinite set has [[cardinality]] [[Aleph number#Aleph-null|aleph-null]] (<math>\aleph_0</math>).
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| *[[Ordinal numbers (linguistics)|Linguistic ordinal numbers]] "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. This can be generalized to [[ordinal number]]s which describe the position of an element in a [[well-ordered]] set in general. An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an [[order isomorphism]] between two well-ordered sets they have the same ordinal number. The first ordinal number that is not a natural number is expressed as <math>\omega</math>; this is also the ordinal number of the set of natural numbers itself.
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| Many well-ordered sets with cardinal number <math>\aleph_0</math> have an ordinal number greater than <math>\omega</math> (the latter is the lowest possible). The least ordinal of cardinality <math>\aleph_0</math> (i.e., the [[Von Neumann cardinal assignment|initial ordinal]]) is <math>\omega</math>.
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| For [[finite set|finite]] well-ordered sets, there is 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]].
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| [[Hypernatural]] numbers are part of a [[non-standard model of arithmetic]] due to [[Skolem]].
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| Other generalizations are discussed in the article on [[number]]s.
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| ==Formal definitions==
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| {{Main|Set-theoretic definition of natural numbers}}
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| Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano axioms state conditions that any successful definition must satisfy. Certain constructions show that, given [[set theory]], [[model theory|models]] of the Peano postulates must exist.
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| ===Peano axioms===
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| {{Main|Peano axioms}}
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| The Peano axioms give a formal theory of the natural numbers. The axioms are:
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| *There is a natural number 0.
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| *Every natural number ''a'' has a natural number successor, denoted by ''S''(''a''). Intuitively, ''S''(''a'') is {{nowrap|''a'' + 1}}.
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| *There is no natural number whose successor is 0.
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| *''S'' is [[injective]], i.e. distinct natural numbers have distinct successors: if {{nowrap|''a'' ≠ ''b''}}, then {{nowrap|''S''(''a'') ≠ ''S''(''b'')}}.
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| *If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of [[mathematical induction]] is valid.)
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| It should be noted that the "0" in the above definition need not correspond to the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are elementarily equivalent in first-order logic, however, there exists models for the Peano axioms which are uncountable; these are called non-standard models for arithmetic and are guaranteed by the Upward Löwenheim-Skolem Theorem. The name "0" is used here for the first element (the term "zeroth element" has been suggested to leave "first element" to "1", "second element" to "2", etc.), which is the only element that is not a successor. For example, the natural numbers starting with 1 also satisfy the axioms, if the symbol 0 is interpreted as the natural number 1, the symbol ''S''(''0'') as the number 2, etc. In fact, in Peano's original formulation, the first natural number ''was'' 1.
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| ===Constructions based on set theory===
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| ====A standard construction====
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| A standard construction in [[set theory]], a special case of the [[von Neumann ordinal]] construction,<ref name="von Neumann1923pp199-208">{{Harvnb|Von Neumann|1923}}</ref> is to define the natural numbers as follows:
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| :Set 0 := { }, the [[empty set]],
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| :and define ''S''(''a'') = ''a'' ∪ {''a''} for every set ''a''. ''S''(''a'') is the successor of ''a'', and ''S'' is called the successor function.
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| :By the [[axiom of infinity]], the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the [[Peano axioms]].
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| :Each natural number is then equal to the set of all natural numbers less than it, so that
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| :*0 = { }
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| :*1 = {0} = {{ }}
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| :*2 = {0, 1} = {0, {0}} = {{ }, {{ }}} | |
| :*3 = {0, 1, 2} = {0, {0}, {0, {0}}} ={{ }, {{ }}, {{ }, {{ }}}}
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| :*''n'' = {0, 1, 2, ..., ''n''−2, ''n''−1} = {0, 1, 2, ..., ''n''−2,} ∪ {''n''−1} = {''n''−1} ∪ (''n''−1) = ''S''(''n''−1)
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| :and so on.
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| When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly ''n'' elements (in the naïve sense) in the set ''n'', and {{nowrap|''n'' ≤ ''m''}} (in the naïve sense) [[if and only if]] ''n'' is a [[subset]] of ''m''.
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| Also, with this definition, different possible interpretations of notations like '''R'''<sup>''n''</sup> (''n-''tuples versus mappings of ''n'' into '''R''') coincide.
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| Even if one [[finitism|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 what it means to be one of these sets. For a set ''n'' to be a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.
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| ====Other constructions====
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| Although the standard construction is useful, it is not the only possible construction. For example:
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| :one could define 0 = { }
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| :and ''S''(''a'') = {''a''},
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| :producing
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| :*0 = { }
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| :*1 = {0} ={{ }}
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| :*2 = {1} = {{{ }}}, etc.
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| :Each natural number is then equal to the set of the natural number preceding it.
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| It is also possible to define 0 = {{ }}
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| :and ''S''(''a'') = ''a'' ∪ {''a''}
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| :producing
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| :*0 = {{ }}
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| :*1 = {{ }, 0} = {{ }, {{ }}}
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| :*2 = {{ }, 0, 1}, etc.
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| The oldest and most "classical" set-theoretic definition of the natural numbers is the definition commonly ascribed to [[Frege]] and [[Bertrand Russell|Russell]] under which each concrete natural number ''n'' is defined as the set of all sets with ''n'' elements.<ref>''[http://www.ac-nancy-metz.fr/enseign/philo/textesph/Frege.pdf Die Grundlagen der Arithmetik:]'' ''eine logisch-mathematische Untersuchung über den Begriff der Zahl'' (1884). Breslau.</ref><ref>Whitehead, Alfred North, and Bertrand Russell. ''[[Principia Mathematica]]'', 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as ''Principia Mathematica to *56'', Cambridge University Press, 1962.</ref> This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of all sets with zero elements) and define ''S''(''A'') (for any set ''A'') as {{nowrap begin}}{''x'' ∪ {''y''} | ''x'' ∈ ''A'' ∧ ''y'' ∉ ''x''}{{nowrap end}} (see [[set-builder notation]]). Then 0 will be the set of all sets with zero elements, {{nowrap|1 {{=}} ''S''(0)}} will be the set of all sets with one element, {{nowrap|2 {{=}} ''S''(1)}} will be the set of all sets with two elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under ''S'' (that is, if the set contains an element ''n'', it also contains ''S''(''n'')). One could also define "finite" independently of the notion of "natural number", and then define natural numbers as equivalence classes of finite sets under the equivalence relation of [[equipollence]]. This definition does not work in the usual systems of [[axiomatic set theory]] because the collections involved are too large (it will not work in any set theory with the [[axiom of separation]]); but it does work in [[New Foundations]] (and in related systems known to be relatively consistent) and in some systems of [[type theory]].
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[Canonical representation of a positive integer]]
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| *[[Countable set]]
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| *[[Integer]]
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| <!--*[[Karatsuba phenomenon]]-->
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{cite book |first=Edmund |last=Landau |authorlink=Edmund Landau |title=Foundations of Analysis |publisher=Chelsea Pub Co |edition=3 |year=1966 |isbn=0-8218-2693-X}}
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| *{{cite book |first=Richard |last=Dedekind |authorlink=Richard Dedekind |title=Essays on the Theory of Numbers |publisher=Dover |year=1963 |isbn=0-486-21010-3}}
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| **{{cite book |first=Richard |last=Dedekind |authorlink=Richard Dedekind |title=Essays on the Theory of Numbers |publisher=Kessinger Publishing, LLC |year=2007 |isbn=0-548-08985-X}}
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| *N.L. Carothers, ''Real analysis'', Cambridge University Press, 2000, ISBN 0-521-49756-6.
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| *Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner. ''Elementary Real Analysis'', ClassicalRealAnalysis.com, 2000, ISBN 0-13-019075-6.
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| *{{Citation|last=Von Neumann|first=Johann|author-link=John von Neumann|year=1923|title=Zur Einführung der trasfiniten Zahlen|journal=Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum|publisher=|pages=199–208|volume=1|url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=}}
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| **{{Citation|last=Von Neumann|first=John|author-link=John von Neumann|editor=Jean van Heijenoort|origyear=1923|date=January 2002|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|chapter=On the introduction of transfinite numbers|edition=3rd|publisher=Harvard University Press|pages=346–354|isbn=0-674-32449-8|url=http://www.hup.harvard.edu/catalog.php?isbn=9780674324497}} - English translation of {{Harvnb|von Neumann|1923}}.
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| ==External links==
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| *{{springer|title=Natural number|id=p/n066090}}
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| *[http://www.apronus.com/provenmath/naturalaxioms.htm Axioms and Construction of Natural Numbers]
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| *[http://www.gutenberg.org/etext/21016 Essays on the Theory of Numbers] by [[Richard Dedekind]] at [[Project Gutenberg]]
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| *{{mathworld|NaturalNumber|Natural Number|}}
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| {{Number Systems}}
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| [[Category:Cardinal numbers]]
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| [[Category:Elementary mathematics]]
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| [[Category:Integers]]
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| [[Category:Number theory]]
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| [[Category:Numbers]]
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| {{Link FA|lmo}}
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