Talk:Natural number/Archive 3

Quotes from pre-algebra books
Pre-algebra is taught in middle school (US grades 6, 7, 8).


 * "Numbers make up the foundation of mathematics. The first numbers people used were the natural or counting numbers, consisting of 1, 2, 3, .... When 0 is added to the set of natural numbers, the set is called the whole numbers." (Chapter 1: Whole Numbers, p. 1)
 * Pre-Algebra DeMYSTiFieD, Second Edition By Allan Bluman (2010)


 * "The most basic collection of numbers is called the natural numbers. The first numbers you learned were probably the natural numbers, those that describe how many objects you can have starting at 1: 1, 2, 3, .... You can have two hands, ten fingers, a dozen cupcakes, one million dollars. All of these quantities are part of the collection of natural numbers. Another important collection of numbers is the whole numbers, the natural numbers together with zero. There are no negatives in the collection of whole numbers." (Chapter 1: The Whole Story, p. 4)
 * The Complete Idiot's Guide to Pre-algebra By Amy F. Szczepanski, Andrew P. Kositsky (2008)

--50.53.36.23 (talk) 10:52, 16 October 2014 (UTC)
 * Fantastic sources User:50.53.36.23. We should be using these as citations for when "counting number" appears.174.3.125.23 (talk) 12:37, 16 October 2014 (UTC)
 * I have added both books to the References section. You may cite them as you like. I have been using harvtxt to generate citations linked to the References. --50.53.36.23 (talk) 18:22, 16 October 2014 (UTC)
 * So "whole number" = "natural number" = "counting number". Are there other synonyms?174.3.125.23 (talk) 12:52, 16 October 2014 (UTC)


 * Both quotes above say that:
 * The "natural numbers" are the set ${ 1, 2, 3, … }$.
 * The "whole numbers" are the set ${ 0, 1, 2, … }$.
 * Notes:
 * Szczepanski & Kositsky use the term "collection" instead of "set" in the quote above.
 * Bluman uses "counting number" once in his book, and that is in the quote above. (per Google books and Amazon searches)
 * Szczepanski & Kositsky say this in their summary of Chapter 1: "Natural numbers are the counting numbers starting with 1 and continuing forever." (p. 17)
 * In their glossary (pp. 313-319), Szczepanski & Kositsky list:
 * "integers All of the natural numbers, their negatives, and zero."
 * "natural numbers ... These are sometimes called the counting numbers: 1, 2, 3 ...."
 * "whole numbers The natural numbers together with zero."
 * They do not list "counting numbers".
 * Conjecture:
 * The term "counting numbers" is obsolescent in mathematics education. (Disclaimer: This is original research.)
 * --50.53.36.23 (talk) 16:41, 16 October 2014 (UTC)

isomorphism of ordered sets
I've tagged this as needing a citation: Halmos and Hamilton don't use the term "isomorphism". Morash lists it in the index, but the Google books snippet doesn't show the indexed page.
 * "... there is a unique isomorphism of ordered sets between them."

What can be used to source the isomorphism?

--50.53.39.110 (talk) 14:07, 15 October 2014 (UTC)

The term "isomorphism" is elementary. It refers to a one-to-one map of one set onto another that preserves certain properties. A group isomorphism preserves group properties. An order isomorphism preserves order properties. In this article, the word is used twice. The fact that it is an "order isomorphism" is stated in one case and implied (but should be stated) in the other. Both of these paragraphs are unclear and need a rewrite. If no one else wants to do it, I'll try to do it. Here is a reference. The Encyclopedic Dictionary of Mathematics, 2nd edition, MIT Press, 1993, ISBN 0262590204, p. 1169 "A mapping $$\phi: A \rightarrow A'$$ of an ordered set A into an ordered set A' is called an order-preserving mapping (monotone mapping or order homomorphism) if $$a \le b$$ always implies $$\phi(a) \le \phi(b)$$. Moreover, if $$\phi$$ is bijective and $$\phi$$ inverse is also an order-preserving mapping from A' onto A, then $$\phi$$ is called an order isomorphism. — Preceding unsigned comment added by Rick Norwood (talk • contribs) 16:19, 15 October 2014‎


 * Thanks for your explanation, the reference, and the extended quote. There is an article on Order isomorphism, and it could use some simplification along the same lines. Anyway, I was actually asking for a source that proves the existence and uniqueness of the isomorphism in the context of the natural numbers. I have modified the tag to make that clear. If the proof is so elementary that it is assigned as an exercise, the exercise would probably be a satisfactory source. --50.53.39.110 (talk) 17:06, 15 October 2014 (UTC)
 * Mendelson (1973) states and proves:
 * Theorem 7.1 Any two Peano systems are isomorphic. (p. 80)
 * Number Systems and the Foundations of Analysis By Elliott Mendelson
 * --50.53.39.110 (talk) 18:26, 15 October 2014 (UTC)
 * Warner (1965) says: "We may now prove that every naturally ordered semigroup is isomorphic to $$( \mathbb{N}, +, \le )$$." (p. 129) He then states and proves Theorem 16.15.
 * Modern Algebra By Seth Warner
 * --50.53.39.110 (talk) 19:35, 15 October 2014 (UTC)

Can these theorems be used to source the statement that says, in part: "... there is a unique order isomorphism between them"? Or is the terminology too different? --50.53.36.23 (talk) 18:37, 16 October 2014 (UTC)

The isomorphism in the Seth Warner quote is an order isomorphism with the additional property that it preserves + (that is f(a+b)=f(a)+f(b). A naturally ordered semigroup is a set which has an identity (usually denoted 0) and an associative binary operation (usually denoted +).  It is naturally ordered if for every element n, if a > b then a + n > b + n.  So, the isomorphism Warner is talking about is an order isomorphism with some additional properties. Rick Norwood (talk) 20:48, 16 October 2014 (UTC)

Proposal to rewrite lede, part 1
I propose the following:

"In mathematics, the natural numbers are those used for counting ('there are six coins on the table') and ordering ('this is the third largest city in the country')."

to

In mathematics, the natural numbers or whole numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country").

and

"The term whole number is also used to refer to the natural numbers, with or without zero. Whole number is sometimes used to refer to any integer, whether positive, zero, or negative. In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers to distinguishing them from the decimal numbers which serve for measuring and often contain a decimal mark."

to

"In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers to distinguishing them from the decimal numbers which serve for measuring and often contain a decimal mark."174.3.125.23 (talk) 15:54, 15 October 2014 (UTC)


 * In the current version, "or" should be "and", because the natural numbers and the whole numbers are used for counting. Also, "or" suggests that the two terms are synonymous:
 * "In mathematics, the natural numbers or and the whole numbers are those used for counting ..."
 * --50.53.36.23 (talk) 04:57, 17 October 2014 (UTC)

Proposal to rewrite lede, part 2
My next proposal is to delete this part of ==History==:

"The term counting number is also used to refer to the natural numbers, with or without zero, though in modern usage it is convenient to use this term to refer to the case where zero is excluded. Some authors use the term whole number to mean a natural number while others use whole number to mean counting number; while still others use whole number to refer to any integer, whether positive, zero, or negative."174.3.125.23 (talk) 16:01, 15 October 2014 (UTC)


 * Could you put  tags around the citations, so they don't get expanded on the talk page? --50.53.39.110 (talk) 16:14, 15 October 2014 (UTC)
 * The citations were being expanded at the bottom of the talk page, which causes confusion and clutter, so I have commented them out by putting them inside wiki markup comments . This doesn't affect the proposed text. --50.53.36.23 (talk) 06:38, 16 October 2014 (UTC)
 * Hi User:50.53.36.23. I saw your request here but was heading to bed so I did not have time to respond, but thanks for doing it for me.  Just to explain why I've proposed these changes is because that the text is being restated, and the later restatement is in ==History== which is seems less germane than if it was in lede.174.3.125.23 (talk) 11:59, 16 October 2014 (UTC)

Proposal of New Lead
I present this for integration so that we don't have edit conflicts on the same section. As has shown in the edit history for the last few days, I have proposed a new lead and there has been some discussion of it in the Discussion of Lead section, but not be the editors here. I am moving the discussion here to this newer section as the editors in this section have been proposing parallel edits. The statements in this proposed lead have been substantiated with citations and discussed in these talk pages. If someone has a argument with citations against a statement in this lead, I hope you will provide a link to that here.

The goal of this proposed lead is to address the successor function issue and construction of arithmetic, i.e. it isn't just for counting, as mentioned in my hatnote comment above, and other threads of discussion on these talk pages. Note comments about counting in the history and etymology of the terminology section as shown there the term counting is ancient beginning with things called counting stones and counting sticks with a continuous history to the present day and should not be written out.

<--> Natural Numbers

The counting numbers are those used for counting objects, {1,2,3, ..} and have their origins in the earliest of mathematics. [link to history section, history of numbers wiki]. In modern times zero is often included in the set of counting numbers so a count can be given when no objects are present to be counted.

In 1899 Peano formalized and abstracted the concept of counting.[Peano citations, abstraction citations] The Peano Axioms give criteria for building a wide variety of sets that may be placed into correspondence with the set of counting numbers. We call such sets Natural Numbers. Examples include the counting numbers, e.g. N={0,1,2 ..}, and many other sets, such as N={{}, –, { – }, ..}, [ link to the Peano section, to the Set-theoretic definition of natural numbers wiki, the transfinite numbers wiki].

Whole numbers are those that can be counted to when starting from 1 and counting by 1, i.e. the same as counting numbers. Should zero be in the domain of discourse it is taken to be a whole number. If negatives are in the domain of discourse, see integer, the negative counting numbers are also taken to be whole. It is sometimes observed that whole numbers are those that do not have a fractional part, but the concept of fraction part belongs to higher level constructions such as rationals or reals, so this observation is not taken as a basis for definition.

Arithmetic is constructed upon the formalization of natural numbers, and Algebra is constructed on top of the formalization of arithmetic.

<--> Thomas Walker Lynch (talk) 06:06, 17 October 2014 (UTC)

I don't think Peano axioms need to be in the lead, and I think the words "wide variety of sets" are misleading. Certainly not every set that obeys the Peano axioms is called the set of natural numbers, that phrase only applies to 0, 1, 2, 3, ... and 1, 2, 3, ... . There are various ways of defining 0 and 1, but that is entirely different from having "a wide variety of sets".
 * The arithmetic structure of the Natural numbers against a successor function is what they are about. Without this structure you have counting numbers. Peano's axioms are number, and order is required to state them. This shift from the naturalist view of counting to the formal view of counting is described in ["History and Philosophy of Modern Mathematics" Minnesota Studies in the Philosophy of Science,  Abrégé d'histoire des mathématiques by Jean Dieudonné, and in "Philosophy of Mathematics and Logic" ed. Shapiro where Arithmetic is referred to as theory of natural numbers]. It is part of the constructions that appeared in many parts of mathematics then or shortly after.
 * The arithmetic structure of natural number sets is the basis of computation theory, where natural number sets are defined such as N ={ "s", "ss", "sss"...} (abstraction based on a first number of "s" and a successor function of appending "s"to a string. [See Papadimitriou "Automata Theory, Languages, and Computation"] Note also the set theoretic N={ {}, –, { –  with the first number of {}, ad the successor function of appending another level of nesting.  This latter set was important in proving such important things as omega being in omega.  Neither of these latter two sets are {0,1,2..}
 * It is a mathematically incorrect to say that the first number in a natural number set is always "0", and the second "1" because these numbers have algebraic properties that are not required in arithmetic.  E.g. The additive identity property of zero is not referenced in the Peano Axioms, and it is not referenced in Arithmetic.  It only occurs two levels up, when one defines an algebraic structure.  I am perfectly fine in defining addition and subtraction on a set that starts with the number 12, for example. As another example, there is no effort made to establish that "{}" has the properties of an additive identity.
 * People do often leave the arithmetic structure implied when using natural number because they don't need to explicitly state it successor, as is done when constructing arithmetic. Though there are many times where this structure is important.
 * I did point out earlier when the convention of zero as a natural number was written out of this article that it was important to mention this convention because 0 is an additive identity for algebraic structures. It does't have to be there, as just discussed, it is just that if algebraic structures do come about it is convenient, see Modern Convention
 * I think these points, and their citations, thoroughly address the questions you have raised. If not, what remains?
 * 218.187.181.237 (talk) 08:25, 18 October 2014 (UTC)

The next paragraph seems much too complicated for the lead.
 * I'll attempt to simplify it.

And the final paragraph does not capture how arithmetic is actually constructed. Arithmetic precedes the formalization by centuries. Rick Norwood (talk) 12:03, 17 October 2014 (UTC)
 * Yes, I would like to add a section with a summary of how arithmetic is constructed from naturals and refer the reader to the arithmetic wiki.
 * No, it doesn't precede. Arithmetic is formally constructed from the first number and successor function of the Peano Axioms, i.e. upon the Natural Number abstraction.  One typically starts the construction by adopting a definition of N= {"s", "ss"...} (quotation marks denoting a subset), then places the successor function in correspondence with the symbol s, - then you can say that repeated successor application is addition.  It goes forward from there.  See [See Papadimitriou "Automata Theory, Languages, and Computation"] for a step by step example of this.  See Shapiro for a universal statement of this, [Shapiro "Philosophy of Mathematics and Logic" page 8], where he notes that arithmetic is the study of the natural numbers.
 * Though modern Arithmetic is constructed from the natural number abstraction (first number and a successor function), yes people have been doing Arithmetic in an informal, and occasionally incorrect, manner since the beginning. The bone in the picture on our article is the earliest known example of math by humands, and believed to have been used for arithmetic as well as counting.  You appeal here to the Naturalists Philosophy of Mathematics, the one that was transformed by Peano's "coup" of providing a construction.  This construction is of central importance to the definition of Natural numbers, but not to counting numbers (or counting sticks, or counting stones, etc.) see also History and Etymology and Origin sections on this talk page.  The observations on other parts of the talk page that counting number is more organic, more human, useful for teaching, only confirms that "counting numbers" is the correct terminology for describing the set Peano worked from when he formalized and abstracted "it".
 * 218.187.181.237 (talk) 08:25, 18 October 2014 (UTC)


 * Oppose the proposal, and in general I oppose anything that gives more prominence to the term whole numbers, which is for the most part not used by mathematicians. We have to accommodate people who use whole number as a search term, but that's all we have to do. --Trovatore (talk) 18:57, 17 October 2014 (UTC)
 * I agree with you. I wish the whole numbers went to the Integer page.  I see it as a digression and artificial appendage here, but there is a long talk section on why they redirect here, and as the redirection is here, whole redirect here I tried to add an explanation.  I did my best to find a short definition, and that "no fractional part" observation is ubiquitous and bared mentioning.  There could be shorter sentence and a section I suppose.  Anway, I don't have strong opinions apart from the one that as the redirect comes here now that something should be said.
 * Notice that reference 8 relative to math education thing and counting numbers is wrong, and should be removed, all Eric said on that page was that it was preferable to talk about integers, than to talk about any of these sets, Counting, Natural, or Whole. If we are to depend on this reference in the lead, it seems all this stuff should redirect to integer.
 * 218.187.181.237 (talk) 08:25, 18 October 2014 (UTC)

unsourced paragraph on notation that was removed from the article
[This unsourced paragraph on notation was removed from the article by here.]

(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, ... }, but rarely in European scientific journals. The notation "$$*$$", however, is standard for nonzero, or rather, invertible elements. The notation $$\mathbb{N}^0$$ could also mean the empty direct product $$\prod_{i=1}^k \mathbb{N}$$ resp. the empty direct sum $$\bigoplus_{i=1}^k \mathbb{N}$$ in the case $$k=0$$.)

--50.53.55.68 (talk) 13:11, 19 October 2014 (UTC)

Is "notion" ambiguous?
A IP user has tagged "notion" as needing clarification, saying that the word is ambiguous. Sure, it may be ambiguous as are almost every English words. Sure "notion of number" could be replaced by "concept of number", but I am not sure that this would clarify anything. Therefore, I'll remove this tag until this IP user will explain which misunderstanding could occur of propose a better term. D.Lazard (talk) 17:04, 19 October 2014 (UTC)
 * "Notion" is used by philosophers, not mathematicians . Can't you find some standard mathematical terminology to use? How about "set"? BTW, do you know about this sense of "notion" in English: Notions (sewing)? See an English-language dictionary for the term "notion". --50.53.55.68 (talk) 17:19, 19 October 2014 (UTC)
 * This footnote has the solution: Say "number systems" instead of "notions of number". The source is Elliott Mendelson. --50.53.55.68 (talk) 17:45, 19 October 2014 (UTC)
 * "Notion" is widely used in mathematics, in the same meaning as in philosophy and the primary meaning of the dictionaries. See, for example Kernel (algebra), Erlangen program, Symmetry, Compact space, Cancellation property, Line (geometry), Projective object, General position, Permutation, ..., where the word appears with this meaning. For more examples, search "notion of" in Wikipedia. "Numbers system" has another meaning, complex number may hardly qualifies as a number system. D.Lazard (talk) 18:45, 19 October 2014 (UTC)
 * All your examples need to be copy-edited. The word "notion" could and should be removed from all of them. The word "notion" is an example of a weasel word. Anyway, a reliable source and a redirect are sufficient reason to use "number systems" instead of "notions of number". --50.53.55.68 (talk) 19:06, 19 October 2014 (UTC)
 * You have just started an edit war. Please admit that you are wrong, and revert. --50.53.55.68 (talk) 19:09, 19 October 2014 (UTC)
 * Bourbaki in English uses "notion" multiple times and "notion of number" three times. (Elements of Mathematics - Algebra part 1) Striking myself. --50.53.47.132 (talk) 04:19, 20 October 2014 (UTC)
 * The French Bourbaki uses "notion" too, so that explains the use in the English Bourbaki: Algèbre: Chapitre 8 --50.53.47.132 (talk) 05:59, 20 October 2014 (UTC)

50.5355.68: You are being silly, and remind me of another editor of this article. Please stop. Rick Norwood (talk) 00:12, 20 October 2014 (UTC)
 * Insisting on accurate and sourced technical terminology is not being "silly". --50.53.47.132 (talk) 00:32, 20 October 2014 (UTC)
 * For the record, James&James say:
 * "NUMBER … number system. … (2) A mathematical system consisting of a set of objects called numbers, a set of axioms, and some operations that act on the numbers, as for the real number system, the complex number system, and Cayley numbers [see CAYLEY—Cayley algebra]."
 * --50.53.47.132 (talk) 00:50, 20 October 2014 (UTC)
 * Firstly there is no reason for insisting to substitute a technical word to a word which is clearly intended as non-technical. Secondly, "number system" does not have any definition that is widely accepted in mathematics (you may hardly find any textbook of algebra or number theory that define and uses this term). Also this term is ambiguous, as almost everybody confuse it with "numeral system". Thirdly "number system", when used, refers to an explicit structured set of number, that is a syntactic object; this is not intended here, where one has to emphasize on the concept (a synonymous of "notion", which IMO would be too pedantic here). D.Lazard (talk) 09:46, 20 October 2014 (UTC)
 * "Thirdly "number system", when used, refers to an explicit structured set of number, that is a syntactic object;"
 * That is essentially what James&James say in sense (2) above. Sense (1) of "number system", which I omitted, appears to be defining what you are calling a "numeral system". There is no separate entry for "numeral system" in James&James. On WP, disambiguation can be done with links, and since there are different articles on the two senses, the ambiguity can be resolved by linking to "number system". NB: I am referring to my copy of the fourth edition of James&James. Do you have a copy of James&James? --50.53.40.231 (talk) 14:22, 20 October 2014 (UTC)

I am going to assume good faith, and explain to 50.53.40.231 why he is wasting our time. Hint, 50.53.40.231, it is not because you insist on accurate and sourced technical terminology, it is because you are nit-picking, and do not understand the sources. To object to "notion" because it is also used in sewing is like objecting to "bat" because it can mean either a baseball bat or a furry mammal. Which meaning is intended is clear from context. You seem not to understand the difference between a number system and a numeral system. A number system is a set of numbers with certain properties, e.g. a group, a ring, or a field. A numeral system is a way of writing numbers, e.g. the decimal system (base 10). You should only edit articles you understand.Rick Norwood (talk) 15:09, 20 October 2014 (UTC)
 * Did you even read my comment? I linked to numeral system and number system, and I referenced both senses as given in James&James. --50.53.40.231 (talk) 16:28, 20 October 2014 (UTC)

equivalent
The "formal definition" section reads as follows: ''The various definitions of the natural numbers are equivalent in the following sense: these sets are naturally ordered and there is a unique order isomorphism[citation needed] between them. Therefore, when using or studying natural numbers, it is not necessary to take care of the particular method that has been used to construct them.'' The first sentence is possibly defensible so long as one sticks to the definition of "equivalence" in terms of the order. Even then one wonders whether this is not merely an equivalence of the ordinal natural numbers. At any rate the second sentence is misleading because it is apparently based on a broader interpretation of equivalence. It can only be claimed that it is unnecessary to pay attention ot the particular method, etc., if the structures are truly equivalent. If there is an aspect which is not equivalent, one cannot make such a sweeping claim of independence of the particular method of definition. For example, Goodstein's theorem is not provable in PA. This is a more careful version of an earlier post that I deleted. Tkuvho (talk) 13:46, 19 October 2014 (UTC)


 * I agree. I've attempted to address this concern. It would be good if you added something about the sense in which PA and set theory are and are not equivalent and mentioned Goodstein's theorem. Rick Norwood (talk) 14:14, 19 October 2014 (UTC)
 * The Formal definitions section no longer mentions any isomorphism theorems. Why did you remove that term? The section says "The two approaches have been proven to be equivalent." That statement will need to be sourced. And the way to say that they are "equivalent" is to cite specific isomorphism theorems. --50.53.55.68 (talk) 16:51, 19 October 2014 (UTC)
 * As I mentioned above, the two approaches cannot be said to be equivalent since there are theorems about the number-theoretic integers that cannot be proved from PA. The most we can affirm here is that certain structures are isomorphic, e.g., the order type as in an earlier version. Misleading blanket statements about the equivalence of the approaches should be avoided. Tkuvho (talk) 17:17, 19 October 2014 (UTC)
 * I cited two isomorphism theorems in the section (one from Mendelson and one from Warner). Could you comment on their relevance to this "equivalence" problem? --50.53.55.68 (talk) 17:26, 19 October 2014 (UTC)

Tkuvho, are you able to fill in the senses in which the two approaches are equivalent, and the ways in which they are not equivalent? Rick Norwood (talk) 00:11, 20 October 2014 (UTC)
 * It would be more appropriate to speak about order-isomorphism (which is sourced, as user 50.53.55.68 pointed out) and avoid making blanket claims about "equivalence", which are not only unsourced but also incorrect, as I already mentioned above (e.g., Goodstein's theorem). Tkuvho (talk) 11:39, 20 October 2014 (UTC)
 * I have edited this paragraph in order to clarify this equivalence. I believe that, now, it may be sourced. I have also edited the whole paragraph for having less ideas in the same sentence, and a clearer explanation (I hope). If this formulation is kept, I am not sure if the sentence between dashes should been kept in the body of the article or moved in a footnote. D.Lazard (talk) 13:39, 20 October 2014 (UTC)
 * There is something odd about claiming a purported "equivalence" between an axiomatic approach, on the one hand, and a specific model constructed by set-theoretic means, on the other. The page should stick to a statement of an order isomorphism as sourced by user 50.53.55.68 and avoid philosophical commitments inherent in such "equivalence" claims. Tkuvho (talk) 06:51, 21 October 2014 (UTC)
 * There is a more immediate problem with the equivalence claim in that it is unsourced and therefore constitutes WP:OR. In fact, I doubt it we can source it since it is incorrect in view of the existence of Goodstein's theorem. Tkuvho (talk) 08:36, 21 October 2014 (UTC)
 * "I believe that, now, it may be sourced."
 * You are going about that backwards. You should be starting with some sources and writing the paragraph based on those sources. --50.53.34.137 (talk) 09:12, 21 October 2014 (UTC)
 * The comments about "truth" in the context of the Peano axioms are really not clear. I think these comments may be confusing the so-called intended interpretation with the Peano axioms.  Tkuvho (talk) 10:29, 21 October 2014 (UTC)
 * IMO, the important fact on which we must emphasize is that the various formalizations are essentially equivalent, sufficiently for allowing the mathematicians, which are not concerned by proof theory, to consider that there only one notion of natural numbers. I am not fully satisfy by the present formalization, because the distinction between true and provable is too technical here, and also controversial. Maybe we could write something like Although the various formalizations are not equivalent from the point of view of proof theory, they are sufficiently close to be equivalent to allow the mathematicians that are not concerned by proof theory of considering that there is only one notion of natural number and not taking care of the chosen formalization. What do you think of this formulation? Can you propose something better? D.Lazard (talk) 13:29, 21 October 2014 (UTC)
 * What sources are you consulting? --50.53.34.137 (talk) 15:23, 21 October 2014 (UTC)
 * Daniel, reassuring mathematicians that "there is only one notion of natural numbers" may be a worthy goal if it is accurate and sourceable. Meanwhile, at Peano axioms we find the following comment: "Peano arithmetic is equiconsistent with several weak systems of set theory.[12] One such system is ZFC with the axiom of infinity replaced by its negation." Tkuvho (talk) 15:45, 21 October 2014 (UTC)

But did you look at Equivalent definitions of mathematical structures? I think, it could be relevant. Boris Tsirelson (talk) 16:31, 21 October 2014 (UTC)

Is there a standard definition of "whole number"?
When some books say one thing and other books say something else, we must report that fact.Rick Norwood (talk) 17:39, 20 October 2014 (UTC)
 * I have added two curriculum standards that define "whole number". The two books and the Ontario source give definitions of "natural number" and "whole number". They happen to be consistent, but I have been careful not to generalize from them. AFAICT, the Common Core State Standards do not mention "natural numbers" or "counting numbers". If you find some reasonably recent pre-algebra books or comprehensive curriculum standards that say something different, please say so. You should be able to find pre-algebra books at a library or a bookstore.


 * Also, note that Szczepanski & Kositsky (2008) say on the inside front cover: "We based this book on the state standards for pre-algebra in California, Florida, New York, and Texas, ..." What curriculum standards are used in Tennessee?
 * --50.53.40.231 (talk) 18:59, 20 October 2014 (UTC)
 * Szczepanski "is a member of the Department of Mathematics at the University of Tennessee". (back cover)
 * Bluman "taught mathematics and statistics in high school, college, and graduate school for 39 years. He received his doctor's degree from the University of Pittsburgh." (p. vii)
 * --50.53.40.231 (talk) 19:46, 20 October 2014 (UTC)
 * Here is a third pre-algebra book that defines the "counting numbers" and the "natural numbers" to be 1, 2, 3, .... The "whole numbers" are defined to be the counting numbers with 0 added to the set. See Chapter 25: Ten Important Number Sets to Know, Counting on Counting (or Natural) Numbers, p. 340:
 * Basic Math and Pre-Algebra For Dummies (2014) By Mark Zegarelli (search for "whole numbers")
 * --50.53.40.231 (talk) 19:17, 20 October 2014 (UTC)

I would like it if it were standard for the natural numbers to begin with 1 and the whole numbers to begin with 0. Unfortunately, many professional mathematicians disagree and disparage the attempt by educators, few of whom are working research mathematicians, to tell research mathematicians what to do. For more information on this subject, see math wars. Rick Norwood (talk) 21:36, 20 October 2014 (UTC)
 * How did you get from complaining about the sources, to explicating a theory about the oppression of research mathematicians by educators? Could you please just comment on the sources per WP:RS? --50.53.49.115 (talk) 22:55, 20 October 2014 (UTC)

Basic Math and Pre-Algebra For Dummies is not as respected a source as, say, Naïve Set Theory by Paul Halmos, but my point is that since sources disagree, and we all accept that sources disagree, adding a large number of sources based on US education "standards" does not change the simple fact that sources disagree. I suppose it doesn't do any harm to expand the list of references, but it doesn't do any good, either. Rick Norwood (talk) 14:05, 21 October 2014 (UTC)
 * Halmos is a reliable source, but you don't need to be told that, right? Both Szczepanski and Bluman have advanced degrees, and their books are published by established publishers. What more can you add about Halmos that makes him more "respected"? Also, both books are based on education standards. The Ontario source is Canadian, not US. Did you even bother to look at it? Would you like me to add some French sources? --50.53.34.137 (talk) 20:32, 21 October 2014 (UTC)

While there is no point in loading the article with a large number of non-noteworthy sources, I'm not the one who reverted your edit. Incidentally, you should know that Paul Halmos is one of the most respected mathematicians of the 20th century. Rick Norwood (talk) 21:41, 21 October 2014 (UTC)


 * A search for "halmos whole number" led to this quote. I hope you like it.
 * "As mentioned earlier, the study of the set of whole numbers, W = {0, 1, 2, 3, 4, ...}, is the foundation of elementary school mathematics." (p. 60)
 * Halmos is quoted on p. 145.
 * Mathematics for elementary teachers: a contemporary approach
 * Gary L. Musser, William F. Burger, Blake E. Peterson
 * J. Wiley, Jan 4, 2005 - Education - 1042 pages
 * --50.53.45.210 (talk) 20:07, 22 October 2014 (UTC)
 * The 10th edition of Musser, et al (2013), references both NCTM and Common Core. (The 10th edition doesn't appear to quote Halmos, but it does reference Pólya several times. Is Pólya respectable enough for you?) --50.53.45.210 (talk) 20:27, 22 October 2014 (UTC)


 * Pólya is good.Rick Norwood (talk) 23:07, 22 October 2014 (UTC)

Bluman's preface
"In his preface, Bluman says that his book is "closely linked to the standard high school and college curricula"

"closely linked" is the problem here. There's a pov problem because Bluman should not be assessing whether or not his book is closely linked or not, since it would simply be subjective.174.3.125.23 (talk) 09:11, 22 October 2014 (UTC)
 * This is a bit of a problem but since this occurs inside a footnote this may be tolerable. Other editors are invited to comment. Tkuvho (talk) 09:38, 22 October 2014 (UTC)
 * I wouldn't consider it tolerable. 3 editors have removed it, you and me included.174.3.125.23 (talk) 09:45, 22 October 2014 (UTC)
 * I was mostly bothered by the registered trademark, which has been removed by now. Certainly if this comment bothers more than one editor it should be re-removed. Tkuvho (talk) 09:48, 22 October 2014 (UTC)
 * The Bluman quote establishes the reliability of the source by connecting the citation to curriculum standards. There is a similar quote from Szczepanski & Kositsky that establishes their reliability. NB: I removed the publisher's trademarked name, because the publisher (McGraw-Hill Professional) is identified in the References, and the publisher also establishes the reliability of the source. --50.53.45.210 (talk) 10:35, 22 October 2014 (UTC)
 * This Bluman quote is quite different from Szczepanski because Bluman is explaining that his book is "closely linked" rather than citing them directly. That's the source of the subjectivity/bias.174.3.125.23 (talk) 11:46, 22 October 2014 (UTC)
 * The phrase "closely linked" is very vague, but it shows that the author and publisher are aware of curriculum standards, and that is sufficient to show that the book seeks to be mainstream, and that it is not fringe. Can you cite something from WP:RS that you believe applies? --50.53.45.210 (talk) 12:02, 22 October 2014 (UTC)
 * I emailed Judy Bass at McGraw-Hill Professional asking what "closely linked" means and for sources (web site or book reviews). --50.53.45.210 (talk) 12:54, 22 October 2014 (UTC)

This was hardly necessary. When an author says (as many do) that his book is "closely linked" with the standards, he means he wrote the book with the standards in front of him and followed them closely.Rick Norwood (talk) 13:44, 22 October 2014 (UTC)
 * This is different from stating "Roger said ...". Bluman's case is "I said, but paraphrased ...".  Additionally, he mentions that they are paraphrased from "... standard high school and college curricula ..." which is impossible if they change every year.174.3.125.23 (talk) 13:58, 22 October 2014 (UTC)

In this context, I think "standard" means NCTM standard, but you are right, for this to be included, it should specify NCTM standard. Rick Norwood (talk) 17:58, 22 October 2014 (UTC)
 * If he did, Allan Bluman should say that. He didn't.174.3.125.23 (talk) 04:37, 23 October 2014 (UTC)
 * I have added the textbook by Musser, et al, which explicitly references both the NCTM and the Common Core standards. IMO, Bluman could be removed as a source, although he is being quoted here. --50.53.45.210 (talk) 09:06, 23 October 2014 (UTC)
 * User:50.53.45.210, thanks for doing this, but over the past couple of days, I've been thinking about this reference you added (Musser et al.) and if either the NCTM and the Common Core standards or both state the use of "natural number" = "whole number", we really should be using the NCTM and the Common Core standards as references, per primary sources. Secondary sources are privy to paraphrasing.  — Preceding unsigned comment added by 174.3.125.23 (talk) 04:10, 26 October 2014 (UTC)


 * An example of the correct reference to use is the ontario reference you used.174.3.125.23 (talk) 04:12, 26 October 2014 (UTC)
 * Actually, secondary sources are preferred, which is why I put Musser, et al, first. See WP:WPNOTRS:
 * "Wikipedia articles should be based mainly on reliable secondary sources, i.e., a document or recording that relates or discusses information originally presented elsewhere."
 * In this context, the Common Core and Ontario standards are primary sources. AFAICT, the NCTM standards are not freely available. The NCTM web site requires payment or registration to access the standards.
 * --50.53.40.60 (talk) 13:02, 26 October 2014 (UTC)

1) Not all standards agree, even in the US.  2)  This article is about the international use of the phrase "natural numbers", not just the US (nor just the US and Canadian) use. 3) All sources indicate that primary and secondary education in the US is extremely poor in math, and therefore should not be used as a standard.  http://www.bbc.com/news/education-20664752  4)  The math taught in primary and secondary schools is only of use to those few students who go on into STEM fields, and in college they must unlearn what they were taught in primary and secondary schools and learn instead the vocabulary of the various professions.

One way in which Wikipedia is used is by American schoolchildren trying to pass standardized tests based on the NCTM or Common Core standards, but that use does not supersede the primary use of Wikipedia as a reference for adults.

Rick Norwood (talk) 11:50, 26 October 2014 (UTC)
 * 'This article is about the international use of the phrase "natural numbers", not just the US (nor just the US and Canadian) use.'
 * That's a good point, except that the article is also about the terms "whole number" and "counting number". And since this is the English WP, it should probably focus on the use of those terms in English. Can you find some relevant sources for Australia, Ireland, New Zealand, or the UK? --50.53.40.60 (talk) 13:37, 26 October 2014 (UTC)
 * There are some statistics in the article on the English-speaking world. --50.53.40.60 (talk) 13:43, 26 October 2014 (UTC)

Both of you are missing the point I am trying to make. Another purpose of Wikipedia is to document the use of a particular term. The terms of "whole number" and "counting number" have been used in reliable sources. Period. Therefore Wikipedia's policy is to include these terms and to describe them. With regard to NCTM or Common Core standards, I only changed the text of the reference User:50.53.40.60 used because it did not indicate what NCTM or Common Core said, but what someone else said about them. I described the situation with the "Roger said ..." example.174.3.125.23 (talk) 03:05, 27 October 2014 (UTC)

Rating change and justification
The description of the B-rating is as follows: "The article has several of the elements described in "start", and most of the material needed for a complete article; all major aspects of the subject are at least mentioned. Nonetheless, it has significant gaps or missing elements or references, needs substantial editing for English language usage and/or clarity, balance of content, or contains other policy problems such as copyright, neutral point of view (NPOV) or no original research (NOR)."

This article clearly meets these standards. I am upgrading the rating to B. With more references, it could be B+ and eventually GA Brirush (talk) 14:47, 9 November 2014 (UTC)

"Grade" vs. "Primary"
Please, accept my assertion that I will not ever touch this nomenclature, especially since I am no profound sage in any school system. I certainly do believe that the notion "Counting numbers" is used throughout the whole educational curriculum, even in examining pupils for its definition, not only in Primary Schools. However, I'd like to point to the fact that Wikipedia redirects from grade school to Primary school and this article just undergoes some (unsourced) debate about the meaning and use ("older Americans"!) of the term grade school, and has been edited just around the reverting of my edit to contain the meaning of "including secondary school". Finally, my revert of the edit relied on the former content, explaining these as synonyma.

Generally, my highly personal stance on this topic is that in schooling there are far to many names, used for only slightly, and unimportantly different entities, in the imho wrong expectation that this diversity helps in understanding important concepts. Peano, in his minimal 5 axioms set, neither required 0 nor 1 to be natural numbers, he just demanded a distinct element (and its successors) to make up the natural numbers, whatever the token(s) to denote it (them) might be. The first is PPOV, the latter a fact. Purgy (talk) 09:52, 23 December 2014 (UTC)
 * I don't think we ought to be delving into what terms are used at which stage in various educational curricula. This is too variable across different educational systems, not interesting enough, and fundamentally, not about the subject of the article, which are the natural numbers.
 * We probably have to say something about the locutions whole number and counting number, but we should discharge this duty in as few words as possible, and move on to the mathematics. This is not an article about terminology. --Trovatore (talk) 19:45, 23 December 2014 (UTC)

Agree with Travatore. Rick Norwood (talk) 19:54, 23 December 2014 (UTC)


 * Hi,, I agree to a greater part to your first sentence, but I also want to make clear that the whole first paragraph of my OP is just my apology for having been involved in some "grade school level" re-reverting of reverted reversions ... I certainly do not want to be a party in some edit war on an article at this here level, where an other articel is edited (174.3.125.23 ) to reason the next step in reverting, and this on a topic I am rather disinterested in and incompetent on.


 * If I were in charge for an improvement of this article, shifting the 0/1 controversy still more to the background, and focussing much more on the uni-directional successor-structure with some arbitrary initial of naturals, which allows just for counting at a first level and a wellordering, came to my mind. The operations of addition, induced by the successor map, which in turn gives rise to a multiplication, were the next steps. I cannot estimate wether and how an attempt of stepping back from the group structures of adding/multiplying and (in consequence) subtracting/dividing is acceptable in hindsight to the targeted audience. Imho, an article on naturals should focus on the properties innate to them and just hint (not too extensively) to the obvious extensions, i.e. adding inverses/neutrals to complete the semigroups to group structures, finally leading to integers and rationals. Just my thoughts. Purgy (talk) 12:29, 26 December 2014 (UTC)

Problems with this article

 * First, most mathematicians use the notation $$\mathbb{N}=\{1,2,3,\ldots\}$$ for natural numbers, unlike this article claims. There are millions of math books and papers using exactly that notation (at least in English or Russian languages).
 * The part of history section about 0 should be moved to the Number article since it has nothing to do with "Natural number". This history part is about whether 0 should be a number or not (it's not about it being a natural number or not).
 * The importance of number 0 is undeniable, but other numbers have similar importance, like $$-1, -2, \frac12, \frac{2}{3}$$, etc. So why 0 should be considered as a natural number and $$-1$$ or $$\frac12$$ shouldn't be then? Even number $$\pi$$ is naturally important. Does that mean that the number $$\pi$$ should be also a natural number?
 * My point is that let's not bring a confusion for an already reserved name "Natural numbers", which for centuries meant (and still means in most books) the numbers $$\{1,2,\ldots\}$$.
 * The set $$\{0,1,\ldots\}$$ already has a name "Whole numbers" or "Nonnegative integers".
 * The term "Counting numbers" is used for the number of elements of a set (thus the term Cardinal numbers is generalization of "Counting numbers"). So counting numbers is actually the same thing as whole numbers, i.e. elements of $$\{0,1,\ldots\}$$. I mean there is no need to change the definition of natural numbers $$\{1,2,3,\ldots\}$$. When the set $$\{0,1,\ldots\}$$ is needed use "whole numbers", "nonnegative integers" or "counting numbers", but don't touch "natural numbers".
 * Peano axioms don't prove anything about including zero to the natural numbers. Actually according to the big math Russian encyclopedia below (p. 228) the same Peano in his first version of axioms used 1 as the first natural number, not 0. In my opinion Peano axioms are redundant in this article, there is no need to repeat them here.
 * Виноградов И.М. (ред.), Математическая энциклопедия, Том 4., Москва, Сов. энциклопедия, 1984.


 * Some other thoughts why the first natural number should be 1:
 * In group theory the order of an element $$a$$ is defined as the least natural number $$n$$ for which $$a^n = e$$, where $$e$$ is the neutral element of a group. That means $$n \in \{1,2,\ldots\}$$, i.e. $$n \neq 0$$, since otherwise every element would have order 0, which would be a pointless definition.
 * Most books starts with Chapter/Section 1, not 0.
 * The first is the same as $$1^{st}$$, not $$0^{th}$$.
 * It would be so unnatural to say that the first natural number is 0.
 * When you start counting something you start with 1, not 0.
 * In Olympic games the gold medal is given to the first place (not $$0^{th}$$ place).
 * $$\mathbb{Q} = \{\frac{p}{q}: p\in\mathbb{Z}, q\in\mathbb{N}\}$$
 * If you split something into parts then the number of parts can only be a natural number. The number 1 means that there is no splitting and it's the smallest number of parts. Number 0 would make no sense here, at least there would be no natural meaning for 0 parts. For example, if we assign empty set for a result of splitting into 0 parts then it would mean that we destroy the amount we had. One might think this way of handling "0 parts" would be OK, but in some sense it contradicts to the next example. Example 2. When we split into 2 equal parts we get $$\frac12$$ of the original size. That way 3 parts give $$\frac{1}{3}$$ of the original size, and splitting into 0 parts gives $$\frac{1}{0}$$ of the original size, which could be interpreted as infinity, or an object of an unlimited mass (which is far from empty set, it's rather the opposite). Anyway, in both ways we are getting very unnatural interpretation of "0 parts" thing. Robertas.Vilkas (talk) 16:47, 13 November 2014 (UTC)


 * You should post at the bottom of the page -- that is where experienced users look for new posts. You should sign your posts with four tildes.


 * You should read the references before you critique the article. If you read the references, you will see references for both 1, 2, 3, ... and for 0, 1, 2, 3, ... .  This article does not change the meaning of natural numbers, it reports the two ways in which the phrase is used.  If you have read much math, you will know that many textbooks do begin with a Chapter Zero, but that is neither here nor there.  Some people define natural numbers one way, some another.  We can't change that.  Neither can you.


 * There is a similar problem with the definition of ring: does a ring have to have a multiplicative identity, or not. Rick Norwood (talk) 13:09, 10 November 2014 (UTC)


 * You've said "many textbooks do begin with a Chapter Zero" and I've said "Most books starts with Chapter/Section 1" and we are both right! Robertas.Vilkas (talk) 17:45, 13 November 2014 (UTC)

This wiki page is so bad that it has inspired a cartoon, so at least some good comes from it: http://www.gocomics.com/saturday-morning-breakfast-cereal/2015/02/16. (The opening of the article confuses the cardinal number set with the natural number set, a fundamental error.) There should be a caveat at the top of the page about errors in the article, less many school children grow up to be bad mathematicians ;-)  — Preceding unsigned comment added by 111.250.116.64 (talk) 22:55, 16 February 2015 (UTC)

I like the cartoon, but could not find any reference there to this article. In any case, let's move this discussion to the bottom of the page, where people look for new posts. Also, I would appreciate it if you signed your posts with four tildes. Rick Norwood (talk) 13:32, 26 February 2015 (UTC)

Fundamental Mathematics Inaccuracies with this article

 * The natural number are not cardinal numbers as stated at least twice in the current article, these sets are of different size.
 * The real numbers can not be constructed from the natural numbers as stated. This is of such fundamental importance to mathematics as to make the current article embarrassing for the current editors.
 * The history section here is irrelevant to natural numbers, which have a very interesting history which sadly is not even touched on here. The current history section does have the flavor of the history of numbers in general but there is already a wiki for that. The leading sentence is even wrong, dots were not the first number representation.  As the picture to the right shows, it was marks on sticks, and counting stones.
 * the primary property of natural numbers is their recursive construction, this has nothing to do with divisibility, as stated in this article, as questions of divisibility are from a higher level algebraic constructs that have division operations.

This article truly needs to be rewritten. U141.211.243.44 (talk) 09:55, 21 February 2015 (UTC)


 * Since there are two recent complaints I boldly but only slightly edited the lede.
 * Up to my knowledge it is common saga that after introducing sufficient quotients of sufficient extensions the reals pop out.
 * No comment on history.
 * I did already mention that imho the successor structure is the core property of the naturals, so I seem to fully agree on this point.
 * Considering the rating and the frequency, this article has its merits too!? Perhaps it is guarded just too insanely jealous by some gatekeepers?


 * Rewriting is definitely beyond my capabilities. Purgy (talk) 14:42, 21 February 2015 (UTC)


 * I read the comments above, and then the article again. I guess the comments about the real numbers has to do with such things as the last digit of the square root of 2 not being odd or even, and the diagnolization of the rationals. Kleene said that the reals could not be constructed without an imprecate assumption.  The statement about Robinson's non-standard analysis and the hyper reals is problem for the same reason.  Robinson explicitly introduces epsilon which is not a natural number.
 * I think this lede is simply trying to say too much in too little space, and is a bit exaggerated in places. How about just chopping a bit out?
 * actually part of the history is there for the naturals, though it is not in the history sections. I will add something to it today.  Perhaps that part could be moved into the current history section, and the current history section could simply be a reference to the history of numbers.
 * the order of infinity problem mentioning cardinal numbers, these too look like symptoms of saying too much. There is a wiki page for transfinite numbers where the nuances missing here can be seen.
 * 162.250.125.163 (talk) 17:29, 24 February 2015 (UTC)


 * Just added a mathematical introduction to Natural Numbers for your review. Most of the material there comes from other sections on this very page.  After just a few seconds the **whole section** disappeared and was removed by DLazard with no specific comments.  If Mr. Lazard has any specific points he feels need further citation or are not of general knowledge I would ask him to add one of these 'needs citation' flags or the like rather than blowing away whole sections of other's work.  This is hardly provocative material.  We aren't discussing the history of Relations between Hindus and Muslims in India or the like. Gee.  — Preceding unsigned comment added by 111.250.103.38 (talk) 17:17, 25 February 2015 (UTC)
 * With this rationale, I can see why 2 people just threw out the section: What you are telling me is that you summarized the page into a new section.  That really is poor writing.  Unfortunately for readers, if a topic is complex, it doesn't mean an article's quality should suffer.174.3.125.23 (talk) 05:18, 26 February 2015 (UTC)

The problem with your post is that it is poorly written. Rick Norwood (talk) 18:08, 25 February 2015 (UTC)


 * This poorly written seems to be a very useful phrase to share edit waring among some similarly thinking conservatives. Properly used it avoids any necessity to discuss content on measures concerning quality, relevance and other properties of content.


 * Seeing some emerging discussions complaining the state of the article, I were less incommunicative. Purgy (talk) 19:26, 25 February 2015 (UTC)

"Similarly thinking conservatives!" You've rattled the wrong cage; I'm more liberal than John Maynard Keynes. But bad writing is bad writing. If anyone really needs a detailed critique, I don't mind giving one, but it takes us far afield from the subject at hand. Rick Norwood (talk) 20:18, 25 February 2015 (UTC)


 * Rick Norwood Sorry you don't like my writing, glad you weren't one of my readers. Please be more specific, anything could be deleted with those words. This is wikipedia, it is a group contribution.  If you feel that a citation is needed for any part, add that citation request to the section in the proper location rather than deleting the entire section.  If there is an incorrect mathematical fact, lets hear it.111.250.103.38 (talk) 02:27, 26 February 2015 (UTC)


 * Rick Norwood: Since based on evidence you are a recidivist on this kind of argueing and since you wrote I don't mind giving one (detailed critique):


 * hic rhodus hic salta!


 * Furthermore, you seem to mix up notions, or are not aware of meaning of "conservative" beyond your own cage, X is X because X is X never has been an acceptable way of proving a claim.


 * Finally, I do not participate in a discussion on this level any more, I restrict myself to express my opinion according to solely my selection in arbitrarily poor written way. Purgy (talk) 07:48, 26 February 2015 (UTC)

I'll be glad to jump to Rhodes for you, but there have been so many changes in the article overnight that it may take a while. I suggest we move this discussion to the bottom of the page.Rick Norwood (talk) 13:27, 26 February 2015 (UTC)


 * Just to have this clear: My last edit here dates from 21.02.2015 and amounts to 63 (sixty three) bytes. Don't mix up locations where your dance is due! Purgy (talk) 20:37, 26 February 2015 (UTC)

Discussion of recent rapid edits.
To begin at the beginning, 111.250.103.38 removed "In mathematics..." with the comment "(the lead speaks of linquistic terms, but starts with "in mathematics", so removed that lead in clause." The language of mathematics and the linguistic definition of mathematical terms is a major part of mathematics.  I'm restoring the phrase "In mathematics" because that is what the entire article, both linguistic and symbolic, is about. Rick Norwood (talk) 13:36, 26 February 2015 (UTC)

Which brings us to the new section titled "In mathematics", which largely repeats what is already in the section "Peano Axioms", placing this technical material higher up in the article.

On the placement of the material: Wikipedia articles on elementary mathematics generally have the more accessible material before the more technical material. On the writing, I'll comment in detail below.

"Due to the Peano Axioms, a set of Natural Numbers is defined to be the smallest set that arises by including as a member a base mathematical entity and then all other entities that would result from applying a non-circular successor function without bound. Such sets are infinite."


 * No, this definition is not "due to" the Peano Axioms. Definitions are not the same as axioms. The rest of the sentence is correct, but too long, and is expressed better in the section titled "Peano Axioms".

"Conventional choices of base entities are the numbers 0, and 1, and the empty set, {}. The corresponding successor functions in these cases are adding 1, or in the set case, concatenating another matched pair of braces. This leads to the following definitions: ℕ0 = {0, 1, 2 …} 	+ 	ℕ1 = {1, 2, 3 …}  	+ 	ℕSet_Theoretic = {{}. – ,{ – } …}"


 * This conflates the Peano axioms and set theoretic constructions. Both are explained in the section "Peano axioms" and are there not conflated.  The word "concatenating" is misused, the notation "ℕSet_Theoretic" is not standard, in the set symbols following there are a couple of minor typos.  A more common construction of the natural numbers using set theory is given in the section "Peano axioms".  It defines the natural numbers as {{},  –, {{}, – },{{}, – ,{{}, – },...}.

"In computation theory it is conventional for the base entity to be a symbol within a computation abstraction, say 's', and the successor function to be defined to append another 's' thus creating strings of symbols. This yields: ℕ = { s, ss, sss, …}"


 * This is fine, but should appear below under "other constructions".

"Set notation alone does not capture the full richness of the definition of Natural Numbers, because members of a set are not ordered, whereas Natural Numbers are ordered. The order among members of a set of Natural Numbers comes from the successor function that was used in the definition. As a consequence of using the successor function the operators less than and greater than, '<', and '>' will also be defined. Furthermore, Arithmetic, i.e. a system with operations of addition and subtraction, follows as repeated successor function applications."


 * This is simply wrong. The set theoretic construction given here can be ordered by number of elements in a maximal sequence of proper subsets.  In other constructions, such as the von Neumann construction, both the successor function and the order relation can be more naturally defined, which is why the von Neumann construction is more common than the one given here, and preferable, since there is no need to endlessly multiply examples.  Finally, subtraction does not follow by repeated successor function applications.

"All of the Natural Number examples given above may be considered to be representational equivalent relative to arithmetic. Accordingly in the computation theory example, 's' may be said to represent '1', 'ss' to represent 2, etc. In the set theoretic example, {} may be said to be a representation for zero, –  a representation for 1 etc. However, relative to other systems there may be differences. For example, an additive identity must be present to satisfy Group (mathematics) properties."


 * The examples given above are not "Natural Number examples" (and Natural Number is not usually capitalized nor are, while we are on the subject, "Arithmetic", "Axiom", or "Set"). The examples given above are examples of different ways to define the set of natural numbers.  "Representational" should be "representationally" since it modifies an adjective, "equivalent", not a noun.  They are not equivalent to arithmetic, they are equivalent to the set of natural numbers, in which arithmetic can then be defined.  There should be a comma after "Accordingly".  The "etc." bypasses a major question about the next element in the series.  Is it to be {{}, – } as above, or is it to be {{}, {{}, – }, as in the von Neumann construction?  Bringing up groups is a red herring, since the natural numbers are not a group, but in the example given above, the natural numbers do have an additive identity, 0, and so why mention sets without an additive identity?

"When a function can be defined which provides a one to one correspondence, also known as a bijection, between the set of Natural Numbers and another set, the Natural Numbers may be used for counting the members of that other set. This is not unique to the set of Natural Numbers, rather it is true in general for sets where one has order in the one set and a bijection leading to another."


 * This is simply wrong. To count the members of a set, you want a bijection between an initial sequence of the natural numbers and that set.  The counting property is unique to the natural numbers in the sense that any set which can be used to "count" finite sets is isomorphic to the natural number.  The counting property is not, however, a property of the real numbers, for example, even though they do have order and can be mapped by a bijection to other sets.

"Related but different sets are the Cardinal Numbers which are used to count the number of elements in a set, and the Ordinal Numbers. As for the natural numbers, arithmetic also exists for these sets. However, both the set of Cardinal Numbers, and the set of Ordinal Numbers are larger than the set of Natural Numbers. All three are infinite. Other related sets include those used in Modular arithmetic (modulo arithmetic), which are finite, and thus smaller than the Natural Numbers."


 * In the penultimate paragraph, it is explained that the natural numbers can be used for counting. This final paragraph says the cardinal numbers are used for counting.  Then, without warning, the linguistic use of "cardinal" and "ordinal" is dropped, and the mathematical use picked up without explanation or introduction.


 * It is a good thing today is a snow day, or I wouldn't have time for this. I'm going to delete the material discussed above, except for the one good paragraph, on computation theory, which I will move to the section on "other constructions".

Rick Norwood (talk) 14:55, 26 February 2015 (UTC)


 * Rick Norwood Not everyone in the world follows your sleep schedule. Have some consideration for others.
 * The two top sections on this talk page brought up problems with the article while you were an editor. You had a chance to make changes but didn't. Now some others have now made some changes of the past few days to contribute and address these issues.
 * Look at the drama here, "in the beginning" "rapid edits" etc. What edits could be more rapid than the ones you just made?  You deleted a whole section and the work of three contributors in one edit.
 * In the comments above you say that the section just duplicates a section you wrote further below. If this is the case, then how bad can it be? Also if not duplicating statements about the Peano axioms is so important then why explain them here when there is already another wikipedia page for them?
 * The other things you mention have fixes other than deleting the section. Others may come along later and address redundancy issues. Wikipedia is a forum of participation, not of exclusion.
 * I am going to undo the deletion of the 'In Mathematics' section, and I challenge you to work edit the material to improve it, or at least find a middle ground with the other contributors.
 * Looks to be a beautiful day here today. We use salt for Margaritas rather than for roads ..
 * 162.250.125.163 (talk) 16:18, 26 February 2015 (UTC)

I'm not the first to delete the section, and I doubt that I'll be the last. I spent several hours pointing out some of the many mistakes in the section. You characterize that as "rapid". I wish! You characterize the more mathematical section near the end of the article as one I wrote. I contributed to it but did not write all of it by any means. How bad can the oft deleted section be? It is full of errors, which I point out. That's how bad it can be. It is not my job, or the job of other editors, to rewrite material that is badly written and unnecessary. You seem not to understand how Wikipedia works, as when you say "Also if not duplicating statements about the Peano axioms is so important then why explain them here when there is already another wikipedia page for them? You should know that it is common for one article to have a brief summary of relevant material, such as the Peano axioms, with a reference to a fuller discussion of that material.  It is not acceptable to repeat material within the same article, except in the lead, which should summarize the salient points.  It is also not acceptable to discuss more advanced topics before more basic topics. Rick Norwood (talk) 16:35, 26 February 2015 (UTC)


 * Rick, this is just more drama. Who are these others?162.250.125.163 (talk) 16:55, 26 February 2015 (UTC)

I've restored the version as of 03:30, 25 February 2015‎, please get consensus here on the talk page before making further edits. Paul August &#9742; 17:07, 26 February 2015 (UTC)


 * Thanks, Paul. The recent edits by multiple IPs pursuing the same agenda and using similar tone may require a WP:SOCK investigation before this gets out of hand (which it may have already).  Tkuvho (talk) 17:28, 26 February 2015 (UTC)


 * I gather from the response from my question is that there were no others who deleted the section. Tkuvho the only agenda I sense is a desire to participate. Note the reversion deletes edits by Woodrow, Prugy, and one IP, so I think this is a fair question as to why a section was deleted rather than edited, and now as to why Paul and others are here requesting consensus - as it limits participation on this article. 162.250.125.163 (talk) 18:12, 26 February 2015 (UTC)
 * D.Lazard reverted here. All edits require consensus see WP:CON, requiring consensus does not limit editorial participation. Paul August &#9742; 18:22, 26 February 2015 (UTC)

Just to have this clear: My last edit here dates from 21.02.2015 and amounts to 63 (sixty three) bytes. I do not feel reverted at all now. This minor edit of mine was caused by more than two reasoned complaints and I hoped to calm the situation of dissatisfaction (I cite: Considering the rating and the frequency, this article has its merits too!?). I stated already elsewhere that I am not interested very much in this article and that I will not take part in a broad discussion (personal attack removed).

In case, I find something I consider of value to remedy this ugly situation, I gladly will suppply it here. (personal attack removed). Best wishes for this article. Purgy (talk) 21:21, 26 February 2015 (UTC)


 * I do not consider the following to be a personal attack, but a necessary component to analyze the process causing the problem and to improve for the future. Of course, I stand by to give evidence of this factual claim:


 * However, I consider it necessary to point to the way how Rick Norwood repeatedly reprimands other editors with unargued "poorly written"-gradings, xxx reverting edits to a state he himself prefers.


 * I left out one single adverb, being itself totally de rigeur, which might possibly be pereceived as aggressive. Purgy (talk) 12:02, 27 February 2015 (UTC)


 * You not quote the terms " (personal attack removed) reverting" and " (personal attack removed) " which are blatant personal attacks. In any case, this page is not the place to discuss someones behavior. If an editor behaves incorrectly (which is not the case for the attacked editor), the right page for complaining is WP:ANI. D.Lazard (talk) 14:12, 27 February 2015 (UTC)


 * As explained on your talk page and mentioned here also I intentionally did not repeat the parts that one potentially could consider a personal attack. I also stated my regret for the part I myself consider to be an attack. I do not understand why you insisted to repeat my fault and I edited out accordingly. Hopefully, I'm not considered a vandal.


 * Finally, I openly confess that I'm guilty of having attacked Rick Norwood personally and I regret this. However, I still do not consider the above factual statement an attack. Purgy (talk) 18:40, 27 February 2015 (UTC)


 * This 'personal attack' banter and 'indignation' posturing is being used to excluded others from contributing.
 * In this last contribution of the "In Math" section, there was never a helpful suggestion, never did there appear any tags in the text or the like. No attempt was made to edit and correct simple mistakes.
 * Instead, new contributions were deleted in total, without notice, with blanket statements, multiple times. Yes, in the "In math" section first by D.Lazard, and then twice by Rick. This was followed by drama.
 * Tkuvho you should follow through with the investigation or put the section back. Maybe it would stop these few established editors from play acting to exclude new contributors. Wikipedia is a community encyclopedia. ..  and do you suppose that these techniques of drama and calls for investigations might be the very reason people come here on IPs?162.250.125.163 (talk) 04:30, 27 February 2015 (UTC)

Hi, Purgy. It may surprise you to know that an IP address does not hide your identity. (You've also slipped a few times in the past, identifying something by an IP address as your own writing. I hope you're having fun.) Rick Norwood (talk) 18:59, 27 February 2015 (UTC)


 * No, I am not surprised. Neither of you, perhaps assuming so, nor of the fact per se. And yes, it happened to me once or twice that I edited some minor remarks when having forgotten to log in, so you know now both, my IP and my Wikipedia nick. I hope these are not at the heart of your fun. Please, in the future refrain from interpreting my behavior, I'll try to do the same with your's. Purgy (talk) 10:20, 28 February 2015 (UTC)