
Natural number can mean either a negative and non-negative numbers integer (, , , , ...) or a non-negative integer (, , , , , ...). Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for partial order ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics. ==History of natural numbers and the status of zero== The natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of numeral system to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful Numeral system#Positional systems in detail system based essentially on the numerals for 1 and 10. The ancient History of Ancient Egypt had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one 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. A much later advance in abstraction was the development of the idea of 0 (number) as a number with its own numeral. A zero numerical digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.#Footnote The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage did not spread beyond Mesoamerica. The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628. Nevertheless, zero was used as a number by all medieval computus (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it. Instead, the Latin word for "nothing," ''nullae'', was employed. The first systematic study of numbers as abstractions (that is, as abstract entity) is usually credited to the ancient Greece philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. In the nineteenth century, a set theory definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. This convention is followed by set theory, logic, and computer science. Other mathematicians, primarily number theory, often prefer to follow the older tradition and exclude zero from the natural numbers. The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers. ==Notation== Mathematicians use N or (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" is added in the latter case: : N = N0 = { 0, 1, 2, ... } ; N* = { 1, 2, ... }. (Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R+ =[0,∞) and Z+ = { 0, 1, 2,... } = N, at least in European literature. The notation "*", however, is quite standard for nonzero or rather invertible elements.)
Less frequently, W or is used for the set of "whole numbers", which are sometimes identified with the natural numbers as defined here, sometimes with the integers (in which case N = W+).
==Formal definitions==
===Peano axioms===
The precise mathematical definition of the natural numbers has not been easy. The Peano postulates state conditions that any successful definition must satisfy:
*There is a natural number 0.
*Every natural number ''a'' has a natural number successor, denoted by ''S''(''a'').
*There is no natural number whose successor is 0.
*Distinct natural numbers have distinct successors: if ''a'' ≠ ''b'', then ''S''(''a'') ≠ ''S''(''b'').
*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.)
It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one).
====The standard construction====
A standard construction in set theory is to define the natural numbers as follows:
:We set 0 := empty set
:and define ''S''(''a'') = ''a'' U {''a''} for all ''a''.
:The set of natural numbers is then ''defined'' to be the intersection of all sets containing 0 which are closed under the successor function.
:Assuming the axiom of infinity, this definition can be shown to satisfy the Peano axioms.
:Each natural number is then equal to the set of natural numbers less than it, so that
:*0 = empty set
:*1 = {0} =
:*2 = {0,1} = {0, {0}} = {{ }, }
:*3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, , {{ }, }}
:and so on. When you see a natural number 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 ''n'' ≤ ''m'' (in the naïve sense) iff ''n'' is a subset of ''m''.
:Also, with this definition, different possible interpretations of notations like R''n'' (''n''-tuples vs. mappings of ''n'' into R) coincide.
====Other constructions====
Although this particular construction is useful, it is not the only possible construction. For example:
:one could define 0 = { }
:and ''S''(''a'') = {''a''},
:producing
:: 0 = { }
:: 1 = {0} = {} , etc.
Or we could even define 0 = {{ }, 0} = {{ }, }
:: 2 = {{ }, 0, 1}, etc.
For the rest of this article, we follow the standard construction described first above.
==Properties==
One can recursively define an Addition in N on the natural numbers by setting ''a'' + 0 = ''a'' and ''a'' + ''S''(''b'') = ''S''(''a'' + ''b'') for all ''a'', ''b''. 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 therefore be embedded in a group (mathematics). The smallest group containing the natural numbers is the integer.
If we define ''S''(0) := 1, then ''S''(''b'') = ''S''(''b'' + 0) = ''b'' + ''S''(0) = ''b'' + 1; i.e. the successor of ''b'' is simply ''b'' + 1.
Analogously, given that addition has been defined, a multiplication × can be defined via ''a'' × 0 = 0 and ''a'' × S(''b'') = (''a'' × ''b'') + ''a''. This turns (N, ×) into a commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Addition and multiplication are compatible, which is expressed in the distributivity:
''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.
If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that ''a'' + 1 = ''S''(''a'') and ''a'' × 1 = ''a''.
For the remainder of the article, we write ''ab'' to indicate the product ''a'' × ''b'', and we also assume the standard order of operations.
Furthermore, one defines a total order on the natural numbers by writing ''a'' ≤ ''b'' if and only if there exists another natural number ''c'' with ''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-order: every non-empty set of natural numbers has a least element.
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 we can find 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 division of ''a'' by ''b''. The numbers ''q'' and ''r'' are uniquely determined by ''a'' and ''b''. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.
==Generalizations==
Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in a ordered sequence and cardinal numbers are used to specify the size of a given set.
For finite sequences or finite sets, both of these properties are embodied in the natural numbers.
Other generalizations are discussed in the article on numbers.
==Footnote==
¹ "... a tablet found at Kish (Sumer) ... thought to date from around 700s_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." [http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html]
Elementary mathematics
Integers
Number theory
Numbers
Set theory
fa:اعداد طبیعی
jbo:Rarna'u
scn:Nummuru naturali
th:จำนวนธรรมชาติ
==Math article protocol== ''Wikipedia follows this convention, as do set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers.'' I have used similar phrasing in other articles, and had it changed, saying "Wikipedia does not refer to itself" or "don't refer to Wikipedia." In mathematics, this would seem to rule out any attempt at universal article terminology protocol. Is this the intended consequence? I'm curious to know, for math articles. User:Revolver 23:50, 9 Jun 2004 (UTC) ---- As Wikipedia is edited by (loosely) the world, it's a little inaccurate to say "the convention used in Wikipedia" in the same way you would say "the convention used in this book", when you neither have checked all the articles in Wikipedia or approve all edits made (or watch all math pages so you can "correct" them). I also see no reason to follow convention arbitrarily set by someone who pretends (s)he can predict the future of Wikipedia (namely, that Wikipedia will include 0 in the natural numbers). Perhaps we should call a vote. --User:Elektron 14:52, 2004 Jun 12 (UTC) ==Peano axioms== I made some substantial changes to the Peano axiom stuff. I hope this makes things more clear, and hopefully, it should also clear up the "debate" about whether 0 is a natural number or not. In essence, we're either talking about the "natural numbers" as a label or term to identify either the set {0, 1, 2,...} or {1, 2, 3, ...} whether you have your set theory or number theory hat on, OR you mean it in terms of Peano axioms. The former sense is already discussed (i.e. the "debate" between what is meant by different people is mentioned) in the first part of the article, so I think the rest should focus on the second. Otherwise, there would be no distinction between "natural numbers" and "countably infinite set"...it's the addition, multiplication, order that we care about! So, this completely changes the "debate" about the status of "zero". The debate that has raged on this talk page largely centers around use of the term to indiciate a set of integers, usage, in other words. That's the less important sense of the term. In the more important sense (Peano axiom sense) the term "zero" simply means any object, that when combined with a successor function, satisfies the axioms. It's like talking about the "zero" of a group. This is why it is perfectly reasonable to take N = {5, 6, 7, ...} "0" = 5 successor of ''a'' = ''a'' + 1 as a system that satisfies the Peano axioms, even though 5 isn't equal to "0" in the normal sense of the number. Of course, once this possibility is mentioned, we can agree to take the usual construction of the natural numbers set-theoretically from that point on. But not doing so introduces problems in logical dependence: *It doesn't make sense to talk about "''a'' + ''1''" as was talked about in the Peano axioms, when the operation of addition hasn't even been defined yet. Nor has the number "1" even been defined yet! You're really talking about the successor function, and skipping ahead of yourself mentally. This confuses things, because it implicitly assumes the existence of N that satisfies the axioms, which is just what you're supposed to be ''proving''. *You might take the usual construction of N and define the order by saying one number is ≤ another if it is a subset of it. Again, this skips ahead of yourself -- the order should only be defined in terms of 0, the successor function, and anything defined in terms of them. This is done correctly here, but it's still a bit confusing earlier when things are said like "the set ''n'' has ''n'' elements"...this is actually a triviality, if you define cardinality in the normal sense -- having ''n'' elements literally means being able to be put in 1-1 corr. with the SET ''n'', also the thing about ordering defined by subsets -- this could also be true by definition, since ordering of ordinals is often defined in terms of the subset relation. User:Revolver 02:58, 13 Mar 2004 (UTC) ==Zero== While I agree with the sentence that many authors have historically excluded 0, would it be possible to add a sentence saying that in this encyclopedia, we always include 0? That way, we can unambiguously use links to :natural number whenever we mean "non-negative integer", an awkward term. --AxelBoldt I disagree. A similar difference of opinion occured here at wikipedia with regard to "ln" versus "log" and it was determined that "ln" should be used because it was unambiguous (although, to my dismay, it was apparently agreed that whenever base 10 was used for "log", it need not be explicitly mentioned or symbolised...as a number theorists, I interpret "log" to mean "natural log" by default, and I have to be reminded if it's otherwise.) So, why not make it a policy to always use the unambiguous terms *positive integer *nonnegative integer (why do we hyphenate, anyway??) *negative integer *nonpositive integer *integer instead of "natural number" or "whole number"? I read a lot of stuff in BOTH set theory AND number theory, and it is the convention in the former to include 0 and in the latter to exclude 0 from the definition of natural number. There are good reasons to justify these decisions in each field -- one definition is not more "correct" than the other, it's just that in set theory, 0 makes conceptual sense to include, while in number theory, it makes similar conceptual sense to exclude it. Thus, ANY choice of definition for natural number is going to go against SOMEONE'S convention. So, why not choose to use the unambiguous terms? I don't really believe they're that "awkward", esp. compared to many other math terms, and more importantly, they're precise and eliminate confusion. BTW, the term "whole number" is rarely used by any mathematicians, in my experience, and I've never seen the symbol "W" used for it, whatever it's intended meaning. user:revolver 14 Jan 2004 Just as a note, I'm an undergrad math student at UC Berkeley and I know I'd get marked incorrectly for including 0 in N. For us, N is {1,2,...} and the set {0,1,2,...} is W. Whether or not this is good or bad is obviously an issue of contention here but I wanted to add what has been institutionalized here. I came here to brush up on mathematical induction but am having some difficulty because I have been taught that induction starts with the base case n=1...I guess it shouldn't be a big difference, or should it? User:Goodralph 04:46, 20 Feb 2004 (UTC) :If you'd get marked wrong for including 0 in N, that probably just means you're taking a number theory or algebra class, not a set theory class. I seriously doubt that Berkeley as a department has some official "policy" that's been institutionalised on the definition on N; if you looked up papers by all faculty members, you'd find papers that use one definition and other the other. As for where to start induction, you can start at any integer you want, 0, 1, -1, -57, 83, etc., since every {''k'', ''k'' + 1, ''k'' + 2,...} is the same as a well-ordered set. User:Revolver 00:24, 21 Feb 2004 (UTC) First, "0" is not 'natural' to start with, since you can't get something from nothing (you can't get 1 from 0). Naturally, we define u0 = 1, and un+1 = un + u0. We only need to have three things already defined: =, 1, +; whereas starting at 0 requires =,0,1,+. I also vote to abandon use of 'natural number' (I've always used it to mean "positive integer", and it's only meant that in all the books I've seen). User:Elektron 05:15, 2004 May 8 (UTC) ==Is "positive integer" also ambiguous?== :I think "positive" is just as ambiguous: does it include zero or not? -- User:Jitse Niesen 14:07, 16 Jan 2004 (UTC) :: "Positive" definitely does NOT include zero. There's no doubt or ambiguity about that, either in English or mathematics. I cannot speak for other languages. User:Peak 07:21, 17 Jan 2004 (UTC) :http://thesaurus.maths.org (mathematical thesaurus, maintained by the University of Cambridge) says "It is not universally agreed whether this set contains zero or not. It is better to use the terms strictly positive and non-negative to indicate whether zero is to be included or not." (see http://thesaurus.maths.org/dictionary/map/word/1011). Indeed, Google finds 55600 pages with "strictly positive", a phrase which would not make sense if there were no doubt whether "positive" includes zero. --User:Jitse Niesen 13:18, 20 Jan 2004 (UTC) [Peak:] Firstly, the page you refer to is about the REAL numbers, not the INTEGERS. Secondly, the quotation you give is incomplete. The first paragraph states unambigously: "The set of positive real numbers ... contains all real numbers greater than zero." (There is no implication here that it might also contain 0, any more than there is the implication that it might contain the negative numbers.) Thirdly, the people in Cambridge are evidently being very polite. Instead of saying, "Some people are confused...", they said "It is not universally agreed..." If some people really insist that the word "positive" means precisely ">=0" I don't really see why they would accept "strictly positive" to mean anything other than "strictly >= 0", using the ordinary meaning of "strictly" (i.e. "without exception"). Fourthly, for Wikipedia, to determine the commonly accepted meaning of a word, it is best to go to good dictionaries such as the American Heritage Dictionary, the online version of which states: : ''11. Mathematics a. Relating to or designating a quantity greater than zero.'' [http://www.bartleby.com/61/41/P0464100.html] Finally, I suspect that you'll find that the frequency of the phrase "strictly positive" has nothing to do with confusion or ambiguity about the meaning of the phrase "positive natural numbers." The first page of Google results that I got had references to real numbers, datatypes, and operators. Even here, the use of the word "strictly" often is for emphasis (i.e. meaning "no exceptions"), as in "He's a strict vegetarian" (i.e. he adheres to the rules strictly). User:Peak 06:39, 21 Jan 2004 (UTC) :I think "strictly" in "strictly positive" refers to the strict inequality ''x'' > 0 as opposed to the inequality ''x'' >= 0. See for instance Bernoulli inequality for this meaning of strict. :However, I do agree that most mathematicians, and most people in general, use "positive" to mean "> 0", and I think Wikipedia should also use it in this sense. The only thing I do not agree with is that it would be clear to everybody that "positive" means "> 0". When I read "positive", I think: this probably means "> 0", but there is a small chance that the author actually meant ">= 0". On the other hand, when I read "natural number", I think: this can either include or exclude zero, and I have to check which definition the author uses if it matters. -- User:Jitse Niesen 12:15, 21 Jan 2004 (UTC) ::Some points to make: ::#I don't know anyone personally who considers 0 to be a positive number. So, just from my own experience in math, considering 0 not to be positive is an almost universal "convention" (although it's not just convention, see next point). ::#The definition of "set of positive elements" of an ordered ring in ring theory specifically excludes 0 as an axiom, for good reasons (essentially, we want to have trichotomy). So, excluding 0 for the integers conforms to this definition. ::#I think the disclaimer above about "no universal agreement" really is a euphemism for "some people are mistaken", or possibly a cavaeat emptor, because of point 1. ::#The use of the word "strictly" is used for inequalities to mean "and not equal", e.g. in a poset, the order relation is usually taken to be "less than or equal to", so you have to explicitly mention that equality is being ruled out. This makes a LOT of difference, esp. in areas like analysis, where the difference between possible equality and strict inequality is crucial. But "positive" already has the strict inequality built into it, since 0 is excluded in the definition. Saying "strictly positive" is not wrong, it kind of emphasises it, but it's not necessary. ::#Most of the google hits had nothing to do with the integers or the real numbers. In other areas of math, they might have their own definition of "strictly positive" (it seems to pop up in Hilbert spaces and operator theory a lot), but that has nothing to do with the integers. ::user:revolver :There's no ambiguity when someone says "12 and under" or "under 12". "positive integer" is also the best way to say "n ≥ 1, n &isa; Z". "strictly positive" means nothing when the reader doesn't know what positive means, just like "integers greater than 0" doesn't mean much when the reader doesn't know the meaning of 'greater than'. I mean, we could say "integers greater than or equal to 1", but who really wants to do that? User:Elektron 05:05, 2004 May 8 (UTC) ==Axioms== "Axioms should be minimal" is a fine statement which doesn't reflect the way mathematicians actually work. For instance, the commonly accepted list of axioms for a vector space is not minimal; nor is the set of axioms for a group. Nevertheless, nearly all sources define groups with the non-minimal and symmetric set of axioms rather the minimal and obscure one. The same is true about the Peano axioms. It's possible to present a minimal version of them; yet these are not Peano's axioms as they're normally presented in mathematical texts. One very good reason to retain the axiom "every natural number except 0 has a predecessor" (which is in fact the commonly accepted form of this axiom) is that it's common to treat the induction axiom as a special and very strong axiom, and to study fragments of Peano arithmetic defined by other axioms without induction, or with weaker forms of it. -- AV I added a note after somebody had modified (vandalized) the first axiom to say "there is not a natural number 0". And there are some that consider 1 the first natural, so I added a note.--User:AstroNomer ---- This system of axioms is too weak, isn't it? To be specific, it allows the construction of what Douglas Hofstadter calls "ω-inconsistency" -- that is, you can define a property X and state that some natural number possesses X, when in fact there is no such natural number. And you still have a consistent system. --User:Juuitchan :Can you (1) prove from the axioms that some natural number possesses X, or can you (2) just not disprove it? I would be very surprised if (1) were the case; on the other hand, (2) is unavoidable and is achieved by several incompleteness results. User:AxelBoldt 04:42 Nov 19, 2002 (UTC) ::I am referring to your (2).
What I want is an axiom that says this: For any natural number ''a'', if you count up like this: 0, 1, 2, 3, etc., by ones, mechanically, like an odometer, you will sooner or later REACH ''a''. I have a feeling that this axiom, while easy to understand, is impossible to formalize, and this is because the axiom depends on the notion of ''time'', which is completely foreign to mathematics.
NB: It might help to understand that I, like many others, rely on pictures as mental models for abstract concepts. My mental model for the natural numbers is as follows: Think of an odometer wheel showing zero. Now this wheel can count forwards and backwards (just not backwards past zero. If it tries that, it will disappear in a puff of smoke.) This is a magical odometer wheel: push it forwards past 9, and it will "grow" another wheel and show 10, and you can keep going, 11, 12, and so on. Push it back past 10, and the extra wheel will disappear. Now, all these positions and forms that the wheels can take correspond to the natural numbers. This is MY model, not some crazy formalization that allows for supernatural numbers.
With my model, the supernaturals are impossible: no matter how far you push the wheel, you will never get a supernatural. --User:Juuitchan ::: Your axiom is equivalent to Peano's last axiom (the axiom of induction). Think about it this way. Obviously 0 is reachable by counting up from 0. Now, if a natural ''n'' can be reached by "counting up" from 0, then ''n'' + 1 can also be reached by "counting up" from 0 -- we just did it. And, therefore, by the axiom of induction, all natural numbers can be reached by "counting up" from 0. ::::No, it isn't. You're assuming that the intuitive property 'can be reached by counting up from 0' can be expressed as a formal proposition, P(n), say, so that the set {n is an element of N: P(n)} can be formed. By the indictinve axiom, this set is all of N. However there is no way to formaulate this intuitive property, other than by the Peano axioms. I.e., we assume that it applies to all n in N. -- User:Daran 11:41, 7 Oct 2003 (UTC) ::::: I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0. So Juuitchan's intuition turns out to be like the parallel postulate in Euclidean geometry. However it appears to be the case that, if you define a set W recursively as { 0 in W; a in W -> S(a) in W } where S(a) is the successor function (+1), then N contains W. In other words, all numbers that can be reached by counting up from 0 are natural numbers, but not necessarily the reverse. ::::: Wikipedia doesn't have a nonstandard arithmetic article. I would write one but I don't know enough about the subject and casual web searches don't turn up enough material. -- User:Zack 19:52, 10 Oct 2003 (UTC) ::: I don't know what you mean by "supernaturals". Infinite ordinals, perhaps? But no one tries to argue that the infinite ordinals are natural numbers. -- User:Zack :You'll never get the entire set either. You model allows for any finite number, however large, but not for the set of all of them. ==Whole number== I put back the text discussing the meaning of the term ''whole number''. This term has a disputed meaning -- I don't question that some people do use it to refer to the integers, but others do use it in the way I originally described. The set letter W is invariably used as I described. I've tried to explain it a bit better this time. An alternative would be to drop this text entirely and discuss the disputed meaning under Whole number. : User:Zack 21:54, 3 Oct 2003 (UTC) I gave the three meanings for "whole number" here for now, without picking sides. I have never seen W for the set of whole numbers, but I'll leave it in. I also removed a non-standard construction of N which will only confuse the reader. User:AxelBoldt 21:26, 6 Oct 2003 (UTC) It looks good. I'm going to do some copyediting in a bit. : User:Zack ==Miscellany== Wouldn't the follow excerpt from this article be more correct of the ''positive integers'' or ''counting numbers''? I agree that zero should be included in the set of natural numbers, but zero is not among the first numbers learned by children, and arguably conceptually more difficult to learn than the counting numbers. :"These are the first numbers learned by children, and the easiest to understand. Natural numbers have two main purposes: they can be used for counting ('there are 3 apples on the table'), or they can be used for ordering ('this is the 3rd largest city in the state')." --User:Sewing 18:28, 18 Dec 2003 (UTC) All said and done, the unfortunate truth is that some sources call zero a natural number and others don't. Tmesipt. 2.20.04. == Better axioms == It's a little stupid to define "natural number" = "positive integer" when you have no definition of positive integer (unless we define "natural number" in terms of the integers). But we can define natural numbers thus: # There is a natural number 1. # For every natural number ''n'', there is a natural-number successor S(''n'') > ''n''. # No other natural numbers exist. # ''n'' + 1 = S(n) # ''a'' + S(''b'') = S(''a'' + ''b''). * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition. * "No other natural numbers exist" iff "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." and is much simpler anyhow. I think it should also imply the predecessor axiom (unless we need another one that says if ''a'' > ''b'' and ''b'' > ''c'' then ''a'' > ''c''). * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞. --User:Elektron 15:08, 2004 Jun 12 (UTC) # For every natural number ''n'', there is a natural-number successor S(''n'') > ''n''. ::What does ">" mean? # No other natural numbers exist. ::Exist besides what? This statement doesn't have meaning for me. It either seems nonsensical or tautological, in neither case is it an axiom. # ''n'' + 1 = S(n) ::Is this a definition or an axiom or what? I don't follow. # ''a'' + S(''b'') = S(''a'' + ''b''). ::Again, is this defining +? You don't have to define + to give axioms for N. * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition. ::Succession isn't defined in terms of addition. It isn't defined in terms of anything. It's just some function satisfying the axioms. Addition has nothing to do with it. * "No other natural numbers exist" iff "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." and is much simpler anyhow. ::The former statement is nonsensical, though. Or too vague. (See above.) The latter is precise. *I think it should also imply the predecessor axiom (unless we need another one that says if ''a'' > ''b'' and ''b'' > ''c'' then ''a'' > ''c''). ::Again, what is ">"? * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞. ::I don't follow. Are you suggesting to change the axioms? They are quite standard and correct as stated. User:Revolver 10:18, 13 Jun 2004 (UTC) ---- I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates). If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (''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''), and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define > or <. "No other natural numbers exist" is an icky way to disallow these. If you do allow use of < in the axioms, then you can say "there is no natural number ''a'' which satisfies ''n'' < ''a'' < S(''n'') for all natural numbers ''n''", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers (in the ''Axioms'' section, ''I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.''), which would not satisfy the mathematical-induction postulate either. I also see nothing wrong with disallowing (implicit or otherwise) definition of, say, {+,>,=} in the axioms. --User:Elektron 07:13, 2004 Jun 18 (UTC) ---- Elektron, I'm not an expert in model theory or anything; my working experience is within the cozy confines of ZFC (really, one can get a ph.d. in math without being exposed to any set theory or logic at all), but I think you're bringing in extraneous issues. Whatever philosophical or model-theoretic issues surround the axioms is worth putting in the article, but this hardly changes the axioms themselves. Whether or not the second-order Peano axioms "capture" what we mean by "the natural numbers" is maybe something for logicians and philosophers to sort through; how does that affect the statement of the (second-order) axioms themselves? ''I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates).'' :I think axiom/postulate are interchangeable terms. I don't know what you mean, "they aren't really axioms". They most certainly are, no different than the axioms defining a group, a topological space, or a geometry. The set of natural numbers with successor function satisfies the axioms, so they're consistent. ''If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (''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''),'' :I'll take you to include "0" in N. And I assume your successor function is defined as "adding one" on {0, 1, 2, ...}, you don't mention how it's defined on the fractions. I don't disagree with what you say here. ''and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define > or <.'' :Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A ∪ FRAC. End of proof. I'm confused, you still seem to be assuming that the "successor function" is synonmous with "add one". It isn't..."successor function" just means "anything that satisfies the axioms". ''"No other natural numbers exist" is an icky way to disallow these.'' :As I said before, the statement "no other natural numbers exist" is nonsensical -- could you please express it more precisely? No other natural numbers exist BESIDES WHAT? I really can't make heads or tails of this statement. ''If you do allow use of < in the axioms,'' :But then, they wouldn't be THE PEANO AXIOMS...they'd be something else. If you want to talk about other sets of axioms besides second-order peano axioms, fine. But it's not a matter of "allowing" <, >, or +, it's just that these aren't what we mean by "second-order Peano axioms", it's like saying "if we do allow * and / (mult and div)" in the axioms for a group...but the group axioms aren't about some hypothetical functions, only about the group operation. Similarly, the second-order peano axioms aren't about ordering or arithmetic operations, only about the SUCCESSOR FUNCTION. ''then you can say "there is no natural number ''a'' which satisfies ''n'' < ''a'' < S(''n'') for all natural numbers ''n''", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers'' :Again, you're ahead of yourself. When discussing the peano axioms, there is no such thing as "the set of natural numbers", this is just a particular example satisfying the axioms. You should be able to state the peano axioms without using the words "natural number". Similarly, I don't know why you feel the need to exclude "infinity", when "infinity" isn't mentioned in the axioms. ''(in the ''Axioms'' section, ''I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.''), which would not satisfy the mathematical-induction postulate either. :"Counting up" is a pretty vague term. In fact, the whole point of the axioms via the successor function is to clarify what "counting up" means. Yes, there are such things as "nonstandard natural numbers" in nonstandard analysis (a la Robinson), but these are not considered to be elements of N, in fact, the definition of a nonstandard natural number is a member of *N not in N. If you're talking about nonstandard models of arithmetic, I talked about this above. There are certainly first-order models of the second-order axioms, such that every first-order statement in the second-order model is provable in the first-order model; this doesn't change the definition of the second-order axioms. User:Revolver 11:39, 18 Jun 2004 (UTC) ---- ''Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A ∪ FRAC. End of proof.'' : That at best proves that A is a set of natural numbers, and says nothing about whether 1/2 is one. User:Elektron 17:59, 2004 Nov 1 (UTC) :Again, you're confusing the axiomatic approach with an "essence" approach. There is no such thing as a \"natural number\". A "natural number" is precisely an element of a structure satisfying the Peano axioms. The question I was answering was not "is 1/2 a natural number?" It was, does the set N = A U FRAC satisfy the Peano axioms, and I showed that is doesn't. (Note that "N" here, isn't N = {0, 1, 2, ...}) We showed that this set N with this particular "successor" function violates the last axiom, because there is a subset A such that A contains 0 and A is closed under the successor function, yet A is a proper subset of N. This is not allowed. Asking the question "is 1/2 a natural number" is nonsensical and jibberish. User:Revolver 03:25, 2 Nov 2004 (UTC) == Hope this clarifies == It seems a lot of the confusion rests on how to interpret the "induction" axiom. In the natural number article, there is an "informal" list of axioms given. This list is just that -- informal, i.e. not precise. There are 2 different ways of interpreting them, if I understand correctly -- # As a set of statements in SECOND-ORDER logic, where the induction axiom takes the usual set-theoretic form of "If a set A satisfies blah, blah, blah, then A = N". THIS is what is usually meant by "Peano axioms". # As an infinite set of statements in FIRST-ORDER logic, where the single induction axiom is replaced by an infinite schema of first-order statements. This axiom schema is also sometimes called "Peano axioms", but most of the time, people mean the former by this term, not this. In any case, whatever you call them, they're not the same. One is a finite set of statements in 2nd-order logic, the other an infinite set of statements in first-order logic. The fact that the former "disallows" natural numbers other than 0, 1, 2, ... and forces a Peano structure to be unique up to isomorphism, and the fact that the latter "allows" nonstandard natural numbers other than 0, 1, 2, .... and allows infinitely many non-isomorphic models of the axioms isn't a contradiction. One simply is not the other. User:Revolver 11:54, 18 Jun 2004 (UTC) == Does place holder mean acceptance as number? == Does the use of the ''numeral'' zero as a placeholder really imply an acknowledgement and true understanding of the ''concept'' of the number zero?? I'm not sure...my first inclination is to say "no". User:Revolver 07:05, 2 Sep 2004 (UTC) :The answer is most certainly no. The "invention" of the zero as a placeholder preceded the "discovery" of zero as a number by many centuries. User:Paul August 02:29, Nov 9, 2004 (UTC) :For a good article on the history of zero, see: [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html] User:Paul August 02:29, Nov 9, 2004 (UTC) ==Circular== Integers are currently defined in terms of natural numbers, and vice-versa. User:24.91.43.225 17:18, 14 Jun 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Natural_number:
Natural_number
Natural_number
Natural_numbers
Natural_number_object
Natural number
Natural number can mean either a negative and non-negative numbers integer (, , , , ...) or a non-negative integer (, , , , , ...). Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for partial order ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics. ==History of natural numbers and the status of zero== The natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of numeral system to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful Numeral system#Positional systems in detail system based essentially on the numerals for 1 and 10. The ancient History of Ancient Egypt had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one 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. A much later advance in abstraction was the development of the idea of 0 (number) as a number with its own numeral. A zero numerical digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.#Footnote The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage did not spread beyond Mesoamerica. The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628. Nevertheless, zero was used as a number by all medieval computus (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it. Instead, the Latin word for "nothing," ''nullae'', was employed. The first systematic study of numbers as abstractions (that is, as abstract entity) is usually credited to the ancient Greece philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. In the nineteenth century, a set theory definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. This convention is followed by set theory, logic, and computer science. Other mathematicians, primarily number theory, often prefer to follow the older tradition and exclude zero from the natural numbers. The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers. ==Notation== Mathematicians use N or (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" is added in the latter case: : N = N0 = { 0, 1, 2, ... } ; N* = { 1, 2, ... }. (Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R+ =
Natural number
==Math article protocol== ''Wikipedia follows this convention, as do set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers.'' I have used similar phrasing in other articles, and had it changed, saying "Wikipedia does not refer to itself" or "don't refer to Wikipedia." In mathematics, this would seem to rule out any attempt at universal article terminology protocol. Is this the intended consequence? I'm curious to know, for math articles. User:Revolver 23:50, 9 Jun 2004 (UTC) ---- As Wikipedia is edited by (loosely) the world, it's a little inaccurate to say "the convention used in Wikipedia" in the same way you would say "the convention used in this book", when you neither have checked all the articles in Wikipedia or approve all edits made (or watch all math pages so you can "correct" them). I also see no reason to follow convention arbitrarily set by someone who pretends (s)he can predict the future of Wikipedia (namely, that Wikipedia will include 0 in the natural numbers). Perhaps we should call a vote. --User:Elektron 14:52, 2004 Jun 12 (UTC) ==Peano axioms== I made some substantial changes to the Peano axiom stuff. I hope this makes things more clear, and hopefully, it should also clear up the "debate" about whether 0 is a natural number or not. In essence, we're either talking about the "natural numbers" as a label or term to identify either the set {0, 1, 2,...} or {1, 2, 3, ...} whether you have your set theory or number theory hat on, OR you mean it in terms of Peano axioms. The former sense is already discussed (i.e. the "debate" between what is meant by different people is mentioned) in the first part of the article, so I think the rest should focus on the second. Otherwise, there would be no distinction between "natural numbers" and "countably infinite set"...it's the addition, multiplication, order that we care about! So, this completely changes the "debate" about the status of "zero". The debate that has raged on this talk page largely centers around use of the term to indiciate a set of integers, usage, in other words. That's the less important sense of the term. In the more important sense (Peano axiom sense) the term "zero" simply means any object, that when combined with a successor function, satisfies the axioms. It's like talking about the "zero" of a group. This is why it is perfectly reasonable to take N = {5, 6, 7, ...} "0" = 5 successor of ''a'' = ''a'' + 1 as a system that satisfies the Peano axioms, even though 5 isn't equal to "0" in the normal sense of the number. Of course, once this possibility is mentioned, we can agree to take the usual construction of the natural numbers set-theoretically from that point on. But not doing so introduces problems in logical dependence: *It doesn't make sense to talk about "''a'' + ''1''" as was talked about in the Peano axioms, when the operation of addition hasn't even been defined yet. Nor has the number "1" even been defined yet! You're really talking about the successor function, and skipping ahead of yourself mentally. This confuses things, because it implicitly assumes the existence of N that satisfies the axioms, which is just what you're supposed to be ''proving''. *You might take the usual construction of N and define the order by saying one number is ≤ another if it is a subset of it. Again, this skips ahead of yourself -- the order should only be defined in terms of 0, the successor function, and anything defined in terms of them. This is done correctly here, but it's still a bit confusing earlier when things are said like "the set ''n'' has ''n'' elements"...this is actually a triviality, if you define cardinality in the normal sense -- having ''n'' elements literally means being able to be put in 1-1 corr. with the SET ''n'', also the thing about ordering defined by subsets -- this could also be true by definition, since ordering of ordinals is often defined in terms of the subset relation. User:Revolver 02:58, 13 Mar 2004 (UTC) ==Zero== While I agree with the sentence that many authors have historically excluded 0, would it be possible to add a sentence saying that in this encyclopedia, we always include 0? That way, we can unambiguously use links to :natural number whenever we mean "non-negative integer", an awkward term. --AxelBoldt I disagree. A similar difference of opinion occured here at wikipedia with regard to "ln" versus "log" and it was determined that "ln" should be used because it was unambiguous (although, to my dismay, it was apparently agreed that whenever base 10 was used for "log", it need not be explicitly mentioned or symbolised...as a number theorists, I interpret "log" to mean "natural log" by default, and I have to be reminded if it's otherwise.) So, why not make it a policy to always use the unambiguous terms *positive integer *nonnegative integer (why do we hyphenate, anyway??) *negative integer *nonpositive integer *integer instead of "natural number" or "whole number"? I read a lot of stuff in BOTH set theory AND number theory, and it is the convention in the former to include 0 and in the latter to exclude 0 from the definition of natural number. There are good reasons to justify these decisions in each field -- one definition is not more "correct" than the other, it's just that in set theory, 0 makes conceptual sense to include, while in number theory, it makes similar conceptual sense to exclude it. Thus, ANY choice of definition for natural number is going to go against SOMEONE'S convention. So, why not choose to use the unambiguous terms? I don't really believe they're that "awkward", esp. compared to many other math terms, and more importantly, they're precise and eliminate confusion. BTW, the term "whole number" is rarely used by any mathematicians, in my experience, and I've never seen the symbol "W" used for it, whatever it's intended meaning. user:revolver 14 Jan 2004 Just as a note, I'm an undergrad math student at UC Berkeley and I know I'd get marked incorrectly for including 0 in N. For us, N is {1,2,...} and the set {0,1,2,...} is W. Whether or not this is good or bad is obviously an issue of contention here but I wanted to add what has been institutionalized here. I came here to brush up on mathematical induction but am having some difficulty because I have been taught that induction starts with the base case n=1...I guess it shouldn't be a big difference, or should it? User:Goodralph 04:46, 20 Feb 2004 (UTC) :If you'd get marked wrong for including 0 in N, that probably just means you're taking a number theory or algebra class, not a set theory class. I seriously doubt that Berkeley as a department has some official "policy" that's been institutionalised on the definition on N; if you looked up papers by all faculty members, you'd find papers that use one definition and other the other. As for where to start induction, you can start at any integer you want, 0, 1, -1, -57, 83, etc., since every {''k'', ''k'' + 1, ''k'' + 2,...} is the same as a well-ordered set. User:Revolver 00:24, 21 Feb 2004 (UTC) First, "0" is not 'natural' to start with, since you can't get something from nothing (you can't get 1 from 0). Naturally, we define u0 = 1, and un+1 = un + u0. We only need to have three things already defined: =, 1, +; whereas starting at 0 requires =,0,1,+. I also vote to abandon use of 'natural number' (I've always used it to mean "positive integer", and it's only meant that in all the books I've seen). User:Elektron 05:15, 2004 May 8 (UTC) ==Is "positive integer" also ambiguous?== :I think "positive" is just as ambiguous: does it include zero or not? -- User:Jitse Niesen 14:07, 16 Jan 2004 (UTC) :: "Positive" definitely does NOT include zero. There's no doubt or ambiguity about that, either in English or mathematics. I cannot speak for other languages. User:Peak 07:21, 17 Jan 2004 (UTC) :http://thesaurus.maths.org (mathematical thesaurus, maintained by the University of Cambridge) says "It is not universally agreed whether this set contains zero or not. It is better to use the terms strictly positive and non-negative to indicate whether zero is to be included or not." (see http://thesaurus.maths.org/dictionary/map/word/1011). Indeed, Google finds 55600 pages with "strictly positive", a phrase which would not make sense if there were no doubt whether "positive" includes zero. --User:Jitse Niesen 13:18, 20 Jan 2004 (UTC) [Peak:] Firstly, the page you refer to is about the REAL numbers, not the INTEGERS. Secondly, the quotation you give is incomplete. The first paragraph states unambigously: "The set of positive real numbers ... contains all real numbers greater than zero." (There is no implication here that it might also contain 0, any more than there is the implication that it might contain the negative numbers.) Thirdly, the people in Cambridge are evidently being very polite. Instead of saying, "Some people are confused...", they said "It is not universally agreed..." If some people really insist that the word "positive" means precisely ">=0" I don't really see why they would accept "strictly positive" to mean anything other than "strictly >= 0", using the ordinary meaning of "strictly" (i.e. "without exception"). Fourthly, for Wikipedia, to determine the commonly accepted meaning of a word, it is best to go to good dictionaries such as the American Heritage Dictionary, the online version of which states: : ''11. Mathematics a. Relating to or designating a quantity greater than zero.'' [http://www.bartleby.com/61/41/P0464100.html] Finally, I suspect that you'll find that the frequency of the phrase "strictly positive" has nothing to do with confusion or ambiguity about the meaning of the phrase "positive natural numbers." The first page of Google results that I got had references to real numbers, datatypes, and operators. Even here, the use of the word "strictly" often is for emphasis (i.e. meaning "no exceptions"), as in "He's a strict vegetarian" (i.e. he adheres to the rules strictly). User:Peak 06:39, 21 Jan 2004 (UTC) :I think "strictly" in "strictly positive" refers to the strict inequality ''x'' > 0 as opposed to the inequality ''x'' >= 0. See for instance Bernoulli inequality for this meaning of strict. :However, I do agree that most mathematicians, and most people in general, use "positive" to mean "> 0", and I think Wikipedia should also use it in this sense. The only thing I do not agree with is that it would be clear to everybody that "positive" means "> 0". When I read "positive", I think: this probably means "> 0", but there is a small chance that the author actually meant ">= 0". On the other hand, when I read "natural number", I think: this can either include or exclude zero, and I have to check which definition the author uses if it matters. -- User:Jitse Niesen 12:15, 21 Jan 2004 (UTC) ::Some points to make: ::#I don't know anyone personally who considers 0 to be a positive number. So, just from my own experience in math, considering 0 not to be positive is an almost universal "convention" (although it's not just convention, see next point). ::#The definition of "set of positive elements" of an ordered ring in ring theory specifically excludes 0 as an axiom, for good reasons (essentially, we want to have trichotomy). So, excluding 0 for the integers conforms to this definition. ::#I think the disclaimer above about "no universal agreement" really is a euphemism for "some people are mistaken", or possibly a cavaeat emptor, because of point 1. ::#The use of the word "strictly" is used for inequalities to mean "and not equal", e.g. in a poset, the order relation is usually taken to be "less than or equal to", so you have to explicitly mention that equality is being ruled out. This makes a LOT of difference, esp. in areas like analysis, where the difference between possible equality and strict inequality is crucial. But "positive" already has the strict inequality built into it, since 0 is excluded in the definition. Saying "strictly positive" is not wrong, it kind of emphasises it, but it's not necessary. ::#Most of the google hits had nothing to do with the integers or the real numbers. In other areas of math, they might have their own definition of "strictly positive" (it seems to pop up in Hilbert spaces and operator theory a lot), but that has nothing to do with the integers. ::user:revolver :There's no ambiguity when someone says "12 and under" or "under 12". "positive integer" is also the best way to say "n ≥ 1, n &isa; Z". "strictly positive" means nothing when the reader doesn't know what positive means, just like "integers greater than 0" doesn't mean much when the reader doesn't know the meaning of 'greater than'. I mean, we could say "integers greater than or equal to 1", but who really wants to do that? User:Elektron 05:05, 2004 May 8 (UTC) ==Axioms== "Axioms should be minimal" is a fine statement which doesn't reflect the way mathematicians actually work. For instance, the commonly accepted list of axioms for a vector space is not minimal; nor is the set of axioms for a group. Nevertheless, nearly all sources define groups with the non-minimal and symmetric set of axioms rather the minimal and obscure one. The same is true about the Peano axioms. It's possible to present a minimal version of them; yet these are not Peano's axioms as they're normally presented in mathematical texts. One very good reason to retain the axiom "every natural number except 0 has a predecessor" (which is in fact the commonly accepted form of this axiom) is that it's common to treat the induction axiom as a special and very strong axiom, and to study fragments of Peano arithmetic defined by other axioms without induction, or with weaker forms of it. -- AV I added a note after somebody had modified (vandalized) the first axiom to say "there is not a natural number 0". And there are some that consider 1 the first natural, so I added a note.--User:AstroNomer ---- This system of axioms is too weak, isn't it? To be specific, it allows the construction of what Douglas Hofstadter calls "ω-inconsistency" -- that is, you can define a property X and state that some natural number possesses X, when in fact there is no such natural number. And you still have a consistent system. --User:Juuitchan :Can you (1) prove from the axioms that some natural number possesses X, or can you (2) just not disprove it? I would be very surprised if (1) were the case; on the other hand, (2) is unavoidable and is achieved by several incompleteness results. User:AxelBoldt 04:42 Nov 19, 2002 (UTC) ::I am referring to your (2).
What I want is an axiom that says this: For any natural number ''a'', if you count up like this: 0, 1, 2, 3, etc., by ones, mechanically, like an odometer, you will sooner or later REACH ''a''. I have a feeling that this axiom, while easy to understand, is impossible to formalize, and this is because the axiom depends on the notion of ''time'', which is completely foreign to mathematics.
NB: It might help to understand that I, like many others, rely on pictures as mental models for abstract concepts. My mental model for the natural numbers is as follows: Think of an odometer wheel showing zero. Now this wheel can count forwards and backwards (just not backwards past zero. If it tries that, it will disappear in a puff of smoke.) This is a magical odometer wheel: push it forwards past 9, and it will "grow" another wheel and show 10, and you can keep going, 11, 12, and so on. Push it back past 10, and the extra wheel will disappear. Now, all these positions and forms that the wheels can take correspond to the natural numbers. This is MY model, not some crazy formalization that allows for supernatural numbers.
With my model, the supernaturals are impossible: no matter how far you push the wheel, you will never get a supernatural. --User:Juuitchan ::: Your axiom is equivalent to Peano's last axiom (the axiom of induction). Think about it this way. Obviously 0 is reachable by counting up from 0. Now, if a natural ''n'' can be reached by "counting up" from 0, then ''n'' + 1 can also be reached by "counting up" from 0 -- we just did it. And, therefore, by the axiom of induction, all natural numbers can be reached by "counting up" from 0. ::::No, it isn't. You're assuming that the intuitive property 'can be reached by counting up from 0' can be expressed as a formal proposition, P(n), say, so that the set {n is an element of N: P(n)} can be formed. By the indictinve axiom, this set is all of N. However there is no way to formaulate this intuitive property, other than by the Peano axioms. I.e., we assume that it applies to all n in N. -- User:Daran 11:41, 7 Oct 2003 (UTC) ::::: I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0. So Juuitchan's intuition turns out to be like the parallel postulate in Euclidean geometry. However it appears to be the case that, if you define a set W recursively as { 0 in W; a in W -> S(a) in W } where S(a) is the successor function (+1), then N contains W. In other words, all numbers that can be reached by counting up from 0 are natural numbers, but not necessarily the reverse. ::::: Wikipedia doesn't have a nonstandard arithmetic article. I would write one but I don't know enough about the subject and casual web searches don't turn up enough material. -- User:Zack 19:52, 10 Oct 2003 (UTC) ::: I don't know what you mean by "supernaturals". Infinite ordinals, perhaps? But no one tries to argue that the infinite ordinals are natural numbers. -- User:Zack :You'll never get the entire set either. You model allows for any finite number, however large, but not for the set of all of them. ==Whole number== I put back the text discussing the meaning of the term ''whole number''. This term has a disputed meaning -- I don't question that some people do use it to refer to the integers, but others do use it in the way I originally described. The set letter W is invariably used as I described. I've tried to explain it a bit better this time. An alternative would be to drop this text entirely and discuss the disputed meaning under Whole number. : User:Zack 21:54, 3 Oct 2003 (UTC) I gave the three meanings for "whole number" here for now, without picking sides. I have never seen W for the set of whole numbers, but I'll leave it in. I also removed a non-standard construction of N which will only confuse the reader. User:AxelBoldt 21:26, 6 Oct 2003 (UTC) It looks good. I'm going to do some copyediting in a bit. : User:Zack ==Miscellany== Wouldn't the follow excerpt from this article be more correct of the ''positive integers'' or ''counting numbers''? I agree that zero should be included in the set of natural numbers, but zero is not among the first numbers learned by children, and arguably conceptually more difficult to learn than the counting numbers. :"These are the first numbers learned by children, and the easiest to understand. Natural numbers have two main purposes: they can be used for counting ('there are 3 apples on the table'), or they can be used for ordering ('this is the 3rd largest city in the state')." --User:Sewing 18:28, 18 Dec 2003 (UTC) All said and done, the unfortunate truth is that some sources call zero a natural number and others don't. Tmesipt. 2.20.04. == Better axioms == It's a little stupid to define "natural number" = "positive integer" when you have no definition of positive integer (unless we define "natural number" in terms of the integers). But we can define natural numbers thus: # There is a natural number 1. # For every natural number ''n'', there is a natural-number successor S(''n'') > ''n''. # No other natural numbers exist. # ''n'' + 1 = S(n) # ''a'' + S(''b'') = S(''a'' + ''b''). * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition. * "No other natural numbers exist" iff "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." and is much simpler anyhow. I think it should also imply the predecessor axiom (unless we need another one that says if ''a'' > ''b'' and ''b'' > ''c'' then ''a'' > ''c''). * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞. --User:Elektron 15:08, 2004 Jun 12 (UTC) # For every natural number ''n'', there is a natural-number successor S(''n'') > ''n''. ::What does ">" mean? # No other natural numbers exist. ::Exist besides what? This statement doesn't have meaning for me. It either seems nonsensical or tautological, in neither case is it an axiom. # ''n'' + 1 = S(n) ::Is this a definition or an axiom or what? I don't follow. # ''a'' + S(''b'') = S(''a'' + ''b''). ::Again, is this defining +? You don't have to define + to give axioms for N. * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition. ::Succession isn't defined in terms of addition. It isn't defined in terms of anything. It's just some function satisfying the axioms. Addition has nothing to do with it. * "No other natural numbers exist" iff "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." and is much simpler anyhow. ::The former statement is nonsensical, though. Or too vague. (See above.) The latter is precise. *I think it should also imply the predecessor axiom (unless we need another one that says if ''a'' > ''b'' and ''b'' > ''c'' then ''a'' > ''c''). ::Again, what is ">"? * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞. ::I don't follow. Are you suggesting to change the axioms? They are quite standard and correct as stated. User:Revolver 10:18, 13 Jun 2004 (UTC) ---- I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates). If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (''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''), and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define > or <. "No other natural numbers exist" is an icky way to disallow these. If you do allow use of < in the axioms, then you can say "there is no natural number ''a'' which satisfies ''n'' < ''a'' < S(''n'') for all natural numbers ''n''", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers (in the ''Axioms'' section, ''I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.''), which would not satisfy the mathematical-induction postulate either. I also see nothing wrong with disallowing (implicit or otherwise) definition of, say, {+,>,=} in the axioms. --User:Elektron 07:13, 2004 Jun 18 (UTC) ---- Elektron, I'm not an expert in model theory or anything; my working experience is within the cozy confines of ZFC (really, one can get a ph.d. in math without being exposed to any set theory or logic at all), but I think you're bringing in extraneous issues. Whatever philosophical or model-theoretic issues surround the axioms is worth putting in the article, but this hardly changes the axioms themselves. Whether or not the second-order Peano axioms "capture" what we mean by "the natural numbers" is maybe something for logicians and philosophers to sort through; how does that affect the statement of the (second-order) axioms themselves? ''I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates).'' :I think axiom/postulate are interchangeable terms. I don't know what you mean, "they aren't really axioms". They most certainly are, no different than the axioms defining a group, a topological space, or a geometry. The set of natural numbers with successor function satisfies the axioms, so they're consistent. ''If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (''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''),'' :I'll take you to include "0" in N. And I assume your successor function is defined as "adding one" on {0, 1, 2, ...}, you don't mention how it's defined on the fractions. I don't disagree with what you say here. ''and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define > or <.'' :Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A ∪ FRAC. End of proof. I'm confused, you still seem to be assuming that the "successor function" is synonmous with "add one". It isn't..."successor function" just means "anything that satisfies the axioms". ''"No other natural numbers exist" is an icky way to disallow these.'' :As I said before, the statement "no other natural numbers exist" is nonsensical -- could you please express it more precisely? No other natural numbers exist BESIDES WHAT? I really can't make heads or tails of this statement. ''If you do allow use of < in the axioms,'' :But then, they wouldn't be THE PEANO AXIOMS...they'd be something else. If you want to talk about other sets of axioms besides second-order peano axioms, fine. But it's not a matter of "allowing" <, >, or +, it's just that these aren't what we mean by "second-order Peano axioms", it's like saying "if we do allow * and / (mult and div)" in the axioms for a group...but the group axioms aren't about some hypothetical functions, only about the group operation. Similarly, the second-order peano axioms aren't about ordering or arithmetic operations, only about the SUCCESSOR FUNCTION. ''then you can say "there is no natural number ''a'' which satisfies ''n'' < ''a'' < S(''n'') for all natural numbers ''n''", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers'' :Again, you're ahead of yourself. When discussing the peano axioms, there is no such thing as "the set of natural numbers", this is just a particular example satisfying the axioms. You should be able to state the peano axioms without using the words "natural number". Similarly, I don't know why you feel the need to exclude "infinity", when "infinity" isn't mentioned in the axioms. ''(in the ''Axioms'' section, ''I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.''), which would not satisfy the mathematical-induction postulate either. :"Counting up" is a pretty vague term. In fact, the whole point of the axioms via the successor function is to clarify what "counting up" means. Yes, there are such things as "nonstandard natural numbers" in nonstandard analysis (a la Robinson), but these are not considered to be elements of N, in fact, the definition of a nonstandard natural number is a member of *N not in N. If you're talking about nonstandard models of arithmetic, I talked about this above. There are certainly first-order models of the second-order axioms, such that every first-order statement in the second-order model is provable in the first-order model; this doesn't change the definition of the second-order axioms. User:Revolver 11:39, 18 Jun 2004 (UTC) ---- ''Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A ∪ FRAC. End of proof.'' : That at best proves that A is a set of natural numbers, and says nothing about whether 1/2 is one. User:Elektron 17:59, 2004 Nov 1 (UTC) :Again, you're confusing the axiomatic approach with an "essence" approach. There is no such thing as a \"natural number\". A "natural number" is precisely an element of a structure satisfying the Peano axioms. The question I was answering was not "is 1/2 a natural number?" It was, does the set N = A U FRAC satisfy the Peano axioms, and I showed that is doesn't. (Note that "N" here, isn't N = {0, 1, 2, ...}) We showed that this set N with this particular "successor" function violates the last axiom, because there is a subset A such that A contains 0 and A is closed under the successor function, yet A is a proper subset of N. This is not allowed. Asking the question "is 1/2 a natural number" is nonsensical and jibberish. User:Revolver 03:25, 2 Nov 2004 (UTC) == Hope this clarifies == It seems a lot of the confusion rests on how to interpret the "induction" axiom. In the natural number article, there is an "informal" list of axioms given. This list is just that -- informal, i.e. not precise. There are 2 different ways of interpreting them, if I understand correctly -- # As a set of statements in SECOND-ORDER logic, where the induction axiom takes the usual set-theoretic form of "If a set A satisfies blah, blah, blah, then A = N". THIS is what is usually meant by "Peano axioms". # As an infinite set of statements in FIRST-ORDER logic, where the single induction axiom is replaced by an infinite schema of first-order statements. This axiom schema is also sometimes called "Peano axioms", but most of the time, people mean the former by this term, not this. In any case, whatever you call them, they're not the same. One is a finite set of statements in 2nd-order logic, the other an infinite set of statements in first-order logic. The fact that the former "disallows" natural numbers other than 0, 1, 2, ... and forces a Peano structure to be unique up to isomorphism, and the fact that the latter "allows" nonstandard natural numbers other than 0, 1, 2, .... and allows infinitely many non-isomorphic models of the axioms isn't a contradiction. One simply is not the other. User:Revolver 11:54, 18 Jun 2004 (UTC) == Does place holder mean acceptance as number? == Does the use of the ''numeral'' zero as a placeholder really imply an acknowledgement and true understanding of the ''concept'' of the number zero?? I'm not sure...my first inclination is to say "no". User:Revolver 07:05, 2 Sep 2004 (UTC) :The answer is most certainly no. The "invention" of the zero as a placeholder preceded the "discovery" of zero as a number by many centuries. User:Paul August 02:29, Nov 9, 2004 (UTC) :For a good article on the history of zero, see: [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html] User:Paul August 02:29, Nov 9, 2004 (UTC) ==Circular== Integers are currently defined in terms of natural numbers, and vice-versa. User:24.91.43.225 17:18, 14 Jun 2005 (UTC)
See other meanings of words starting from letter:
N
NA | NB | NC | ND | NE | NF | NG | NH | NI | NJ | NK | NL | NM | NO | NP | NR | NS | NT | NU | NW | NX | NY | NZ |Words begining with Natural_number:
Natural_number
Natural_number
Natural_numbers
Natural_number_object
These materials are based on Wikipedia and licensed under the GNU FDL
YouTube.com videos better site than Turbo Tax 2007
