Talk:Natural number/Archive 1

message: 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. Revolver 23:50, 9 Jun 2004 (UTC)
thread_title: Math article protocol

message: 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.
Revolver 02:58, 13 Mar 2004 (UTC)
thread_title: Peano axioms

message: 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')."
--Sewing 18:28, 18 Dec 2003 (UTC)
thread_title: Miscellany

message: 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.
Revolver 11:54, 18 Jun 2004 (UTC)
thread_title:  Hope this clarifies 

message: Integers are currently defined in terms of natural numbers, and vice-versa. 24.91.43.225 17:18, 14 Jun 2005 (UTC)
thread_title: Circular

message: I am always confused why number theorists do not include 0 in N, since without it, there is no additive identity.
Though it was not "discovered" until numbers had been in use for a long time, I think zero is quite "natural" for counting:  I can imagine having 1, 2, or 0 objects, while -3 or 2 1/2 require some kind of logical "leap."  I guess this explains why set theorists like 0 in N, since the cardinality of a finite set is always in N.
:I don't know why number theorists don't include 0 (granting that they don't; I haven't talked to one about it recently). It's just their convention, I suppose, and doesn't really need a justification. All we can really do in an encyclopedia is report both conventions, which we correctly do, and I would like to suggest that detailed discussion of it take place in some other forum, because it's not adding value to the article to keep rehashing it. --Trovatore 17:03, 1 November 2005 (UTC)
thread_title:  Identity 

message: I added a description of the Frege-Russell definition of the natural numbers (which
works in New Foundations and related systems) to the "Other constructions" section.
Randall Holmes 21:33, 19 December 2005 (UTC)
thread_title:  Reference to Frege-Russell definition added to "Other constructions" 

message: I don't understand why there's even an argument. If you look inside any decent college math textbook that defines the natural numbers, you will see a definition that says, "the set of natural numbers is the union of all inductive sets." Your arguments about including 0 is pointless because that's not what the natural numbers are about. The most important thing about the natural numbers is that it's an inductive set. Moreover, it's THE inductive set. —The preceding unsigned comment was added by Davexia (talk • contribs)  21:58, 7 July 2006   (UTC)
: There is already a section on formal defintion in the article. I think that's enough and there is no need to add in the formal definition in the intro. Also, is it correct that
:: the set of natural numbers is the union of all inductive sets
: as you state? By the way, I found mathworld's iductive set clearer than our inductive set. Oleg Alexandrov (talk) 00:04, 8 July 2006 (UTC)
replies: text: Our inductive set is about a concept from descriptive set theory. The article corresponding to the Mathworld article is at inductive set (axiom of infinity). The latter concept is not one that really needs to be named, and I don't think it ordinarily is named; people just talk about "a set closed under successor".  --Trovatore 00:33, 8 July 2006 (UTC)
replies: text:  OK, then it is wrong to say that "the set of natural numbers is the union of all inductive sets". Rather, N is the smallest set which is closed under suscessor, up to isomorphism. Oleg Alexandrov (talk) 02:18, 8 July 2006 (UTC)

text: "Inductive set" sounds like it should have a definite meaning, but I am not aware of any standard definition. Sometimes "hereditary set" is used for the same kind of thing, but that also has multiple meanings. JRSpriggs 04:58, 8 July 2006 (UTC)
thread_title: Formal Definition

message: I am confused by the definition
:\sigma(A) (for any set A) as \{x \cup \{y\} \mid x \in A \wedge y \not\in x\}
Should that not be more like:
:\sigma(A) (for any set A) as \{\{x \cup \{y\} \mid x \in A \} \mid \forall y \not\in x\}
As it stands it does not appear to define a set of sets. My maths is a bit rusty, so please comment. −Woodstone 12:22, 8 July 2006 (UTC)
replies: text: The formula in the article would be a correct way of defining successor that one could use, if there were a set of all n-element sets which we could call "n". (σ is being used here for the successor function.) If A were "n" so defined, then x would be an n-element set and y would be a potential n+1st element to be added to the set. Does it make sense now? JRSpriggs 08:00, 9 July 2006 (UTC)

text: Assuming that A IS the set of all n-element sets, then \sigma(A) as defined in the article is the set of all n+1-element sets. Perhaps you do not understand the notation. In the expression \{x \cup \{y\} \mid x \in A \wedge y \not\in x\}, the part on the left, x \cup \{y\}, is to be included in the set for EVERY x and EVERY y that statisfy the condition on the right, x \in A \wedge y \not\in x. Does that help? JRSpriggs 09:25, 9 July 2006 (UTC)

text: You cannot correctly include such quantifiers in this notation for sets using braces because you cannot apply the scope of the quantifier to the correct stuff. If you want to be so explicit, then you must abandon the braces and write down a formula such as \forall A \forall s(s \in \sigma(A) \iff \exist x \exist y[s = (x \cup \{y\}) \land x \in A \land y \notin x]). This implies that \forall A \forall x \forall y((x \cup \{y\}) \in \sigma(A) \Leftarrow [x \in A \land y \notin x]). JRSpriggs 01:40, 10 July 2006 (UTC)
thread_title:  Confused 

message: This article appears in need of improvement both in form and in content. I touched it, barely, to clean up a little formatting, but have not given it anywhere near the attention it should get. In particular, I am explicitly not vouching for the contents, which may be gibberish or genius (though I doubt either extreme). I'd also consider adding a mention of "natural number objects" in category theory. --KSmrqT 04:56, 9 July 2006 (UTC)
thread_title:  Disclaimer 

message: I have a question elicited by the fargoing abstractions in the article. With all this abstaction it looks to me like any countable set is defined to be the natural numbers. That would imply that the rational numbers are equal to the natural numbers. Correct or not? If so: didn't you go too far? If not: why are they different? −Woodstone 09:46, 9 July 2006 (UTC)
replies: text: Clearly, using a non-standard model of the natural numbers would cause a great deal of confusion so it should be avoided whenever possible. If none-the-less you use a non-standard model, then you must specify not just the base set (which could be any countable set as you said), but also which element plays the role of zero and what function plays the role of successor. With such additional information, the rational numbers used as the base set for a model of the natural numbers would be distinguishible from the rational numbers regarded as usual, as the smallest field extending the integers. JRSpriggs 10:14, 9 July 2006 (UTC)

text: So what is your point? That you can devise something which is confusing? And then be confused by it? JRSpriggs 01:45, 10 July 2006 (UTC)

text: Yeah, the rationals are in bijection with the naturals.  The next question is: so what?  Remember that the naturals and the rationals are both monoids, and that they are not isomorphic as monoids.  They are both totally ordered, but not isomorphic as ordered sets.  It's only when you forget about all extra structures that you're allowed to view them as the same.  -lethe talk [ +] 02:23, 10 July 2006 (UTC)
thread_title:  Countable sets 

message: Substituted at 21:45, 3 May 2016 (UTC)
thread_title: Assessment comment