Talk:Natural number

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class Top Priority  Field: Number theory
A vital article.
One of the 500 most frequently viewed mathematics articles.
This article has been reviewed by the Version 1.0 Editorial Team.

Archives

Contents

[edit] A successor

Maybe the Peano axiom "Every natural number a has a natural number successor, denoted by S(a)." should rather read "Every natural number a has exactly one natural number successor, denoted by S(a)." I know, that the last (mathematical induction) axiom implies this, but the proposed form (in contrast to the current form) is in accord with the use of the term successor here ("the successor").

Or change the use of the term successor so that it is not silently assumed that the successor is only one, could be even better. (something like a+1 instead of S(a) and "if a property is possessed by 'some' successor" instead of "... possessed by the successor" in the last axiom) --trosos 213.220.249.112 15:55, 13 January 2007 (UTC)

[edit] |N

I've seen the set |N discussed in the sci.math newsgroup occasionally; is this another notation for N, or does it have some other special meaning? Or is this an attempt to represent a special character within the limitations of ASCII? Should it be mentioned in the article? I found this informal definition in a newsgroup post:

The informal definition |N = {1,2,3,...} is usually taken to mean that |N is a set S such that
(1) 1 is a member of S
(2) for each member n of S, n+1 is also a member of S, and
(3) |N is a subset of every set, S, with properties (1) and (2).

I'm not clear on (3), except that I think it means that S can be any set containing consecutive naturals (and possibly other members as well), and therefore |N is a subset of any such set. — Loadmaster 23:13, 9 April 2007 (UTC)

It's an attempt to mimic blackboard bold in ASCII. It's almost always more confusion than it's worth. No, I don't think it deserves mention in the article. --Trovatore 23:16, 9 April 2007 (UTC)

[edit] Definition

Couldn't the first sentence just be "A natural number is a number greater than zero with no decimal separator"? For people who just want to know the basic definition without reading the entire article? —Preceding unsigned comment added by 86.76.137.45 (talk)

No. That is a definition by non-essentials. You should not confuse a number with a particular representation of that number (in this case the decimal representation). JRSpriggs 10:38, 22 April 2007 (UTC)

[edit] Frege

What Frege defined in "Die Grundlagen der Arithmetik" were not natural numbers but cardinal numbers as one figures out by reflecting -- for example -- upon the set of all sets with #\mathbb{N} elements. Such a set meets the definition but sure is no natural. —The preceding unsigned comment was added by 195.176.59.181 (talk) 09:01, August 20, 2007 (UTC)

[edit] Peano axioms and isomorphism

The section on the Peano axioms claims that "All systems that satisfy these axioms are isomorphic". This would seem to contradict both the incompleteness theorem and the Löwenheim–Skolem theorem. 72.75.107.59 (talk) 01:29, 19 January 2008 (UTC)

Those refer to first-order logic. What the claim means is that all structures that satisfy the full Peano axioms, in the sense of second-order logic, are isomorphic. --Trovatore (talk) 01:33, 19 January 2008 (UTC)
Aren't the axioms listed there ("these axioms") all first-order? 72.75.107.59 (talk) 01:40, 19 January 2008 (UTC)
No, the full axiom of induction is not first-order. It becomes a first-order axiom schema if you limit the properties being considered to ones that can be defined by a first-order formula. --Trovatore (talk) 01:56, 19 January 2008 (UTC)

[edit] Number 4

this is topical: Mathematics to Retire Number 4--Billymac00 (talk) 14:58, 4 April 2008 (UTC)

This is funny, but I don't think it should be in the article. Oleg Alexandrov (talk) 15:24, 4 April 2008 (UTC)

[edit] About zero as natural number

In the article I read "To be unambiguous about whether zero is included or not, sometimes an index "0" is added ..." The funny thing is this isn't unambiguous as in my country (Belgium), zero is a natural number, and this symbol \mathbb{N}_0 means without zero, because \mathbb{N} represents the natural numbers, which is with zero. Also for other number sets a sub-index 0 means "without zero" (in Belgium). Something extra I forgot first to mention: some people may wonder why things like \mathbb{Z}^+ includes zero, but zero is in Belgium a positive AND a negative number so it is logic that in that case things like \mathbb{Z}^+ means the positive numbers, which is with zero. In other countries in Europe is this not always the case, it depends.193.190.253.144 (talk) 21:47, 7 June 2008 (UTC)

Interesting -- it strikes me as really counterintuitive to write \mathbb{N}_0 for the specific purpose of excluding zero. However I also have not come across this notation as a way of allowing zero, which is what the article currently claims -- can anyone find this attested somewhere? --Trovatore (talk) 22:49, 7 June 2008 (UTC)
193's statement is at least partially confirmed by the Dutch Wikipedia. In the article Natuurlijk getal (Natural number) the notations \Z^+ and \Z^- are given as including 0, while \Z_0^+ and \Z_0^- are given as excluding 0. Also Positief getal (Positive number), Negatief getal (Negative number), and 0 (getal) (0 (number)) state that in Belgium the number 0 is considered both positive and negative. Dutch is one of the official languages of Belgium. The French Wikipedia is equivocal; for example, it defines Nombre positif (Positive number) as: un nombre qui est supérieur (supérieur ou égal) à zéro ("a number that is greater than (greater than or equal to) zero"), without explaining when the parenthesis is supposed to be in effect. Next thing, the French article states: Zéro est un nombre réel positif, et est un entier naturel. Lorsqu'un nombre est positif et non nul, il est dit strictement positif. ("Zero is a real positive number, and is a natural number. When a number is positive and non-zero, it is called strictly positive.") The Spanish Wikipedia, at Número positivo, has a similar text but is slightly clearer in expressing that the meaning is ambiguous. In Germany and Italy the number 0 is unequivocally neither positive nor negative.  --Lambiam 07:56, 8 June 2008 (UTC)