Talk:Set-theoretic definition of natural numbers

Introductory gif
I really don't think the gif in the first section contributes anything. It's just confusing. — Preceding unsigned comment added by 68.232.122.172 (talk) 20:12, 10 October 2016 (UTC)


 * I agree. &mdash; Carl (CBM · talk) 00:05, 11 October 2016 (UTC)

Definition per Frege and Russell
I added a discussion of the definition of Frege and Russell which works in naive set theory, type theory, and New Foundations.

Randall Holmes 18:02, 15 December 2005 (UTC)


 * It would be interesting to discuss the relation of this definition to logicism. Tkuvho (talk) 10:13, 20 October 2010 (UTC)


 * What does it mean for a definition to “work” in a theory known to be inconsistent such as naive set theory? 2604:6000:B405:F200:45EC:9BBE:F1C5:A9D9 (talk) 16:01, 12 February 2017 (UTC)

The definition of natural numbers as finite cardinals (P.Suppes) could be added, for example with the following wording:

« The equivalence classes have to be replaced by cardinals. The simplest way to introduce cardinals is to add a primitive notion,  Card, and an axiom of cardinality to ZF set theory (without axiom of choice).

Axiom of cardinality (A.Fraenkel): The sets A and B are equinumerous if and only if Card(A) = Card(B)

Definition: the sum of cardinals K and L such as K= Card(A) and L = Card(B) where the sets A and B are disjoined, is Card (A ∪ B).

The definition of a finite set is given independently of natural numbers: a set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion.

Definition: a cardinal n is a natural number if and only if there exists a finite set x such that n = Card(x)

0 = Card (∅) 1 =Card({A})

Definition: the successor of a cardinal K is the cardinal K + 1

Theorem: the natural numbers satisfy Peano’s axioms. » CBerlioz (talk) 08:16, 22 September 2022 (UTC)

This entry needs more and better content
Randall, I've given this article my usual polish but remain dissatisfied with it. The way you set out the Frege-Russell definition catches me off-guard. I do not have a sense that you have clearly demarcated Frege-Russell's way from Von Neumann's way. You did not mention that you were defining the Von Neumann ordinals, and that extracting the corresponding cardinals from these ordinals requires Choice. (The entry says this now, but only because I added it.) I am not confident that the two definitions in this article are killingly accurate, and maximal accuracy is important here, otherwise a poor reader might take Julius Caesar for a number, as per Frege's notorious worry! I'm adding a link to your wonderful 1998 text.

As I have told you before via another channel, the Frege-Russell way has always struck me as "just right." The fact that the resulting equivalence classes are not sets in ZFC does not damn Frege-Russell; rather it damns ZFC. A foundational system of mathematics should come with an elementary and intuitive definition of the finite cardinals, period. ZFC fails that test, NFU does not.202.36.179.65 18:13, 16 July 2006 (UTC)

Small edit
The description of the empty set in bracketts after the empty set symbol was a) wrong and b) unnecessary, so i replaced it with a link to the empty set page. I don't mean the description was incorrect it was just very poor english, and confusingly similar to saying "the set containing zero" which is of course incorrect. Triangl 01:12, 12 November 2006 (UTC)

Non-equivalence of "the empty set" and the symbol used before "(the empty set)"
The symbol used to describe the empty set is incorrect. {} denotes the empty set as does the symbol Ø. Then {Ø} is truly an non-empty set, because it is a set that contains the empty set. To formally define zero to be {Ø} might be true, but placing a link to the empty set after {Ø} is misleading. bradskins 22:21, 27 November 2006 (UTC)


 * I changed it, and you could have changed it as well. CMummert 11:54, 28 November 2006 (UTC)


 * It would be interesting to know where the symbolism { } came from. Granted that it seems now to be an obvious extension of "empty" and "set", Zermelo 1908 and Fraenkel 1922 used "0" and worked from there. von Neumann 1925 used O (as in "oh", not the zero-symbol). wvbaileyWvbailey 14:59, 8 October 2007 (UTC)

Am confused about origin of the axiom of infinity: the notion of adjoining the previous to a set of the previous
The following is quoted from the Peano axioms page:
 * The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:
 * s(a) = a ∪ { a }.

Here I too had thought that von Neumann was the originator of this idea of, beginning with 0 the empty set 0, adjoining the set {0} to 0 to create the next element, and etc, but then I read this:

Zermelo 1908:
 * “AXIOM VII. (Axiom of infinity <>). There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as an element.” ((Zermelo 1908 in van Heijenoort 1967:204)

Am I reading this correctly as Z = { 0, {0,{0}} =def1, {1,{1}} =def2, ..., etc }? What did von Neumann 1925 bring new to the party? (I have more quotes from Fraenkel and von Neumann if anyone's curious). At least relative to these definitions above and this article and Peano Axioms, I cannot see what von Neumann did that was new, and it would seem that rather than von Neumann be cited, Zermelo 1908 should be. Lemme know, Thanks, Bill Wvbailey 14:52, 8 October 2007 (UTC)


 * I don't know about the historical origins of von Neumann ordinals, but I want to point out that the infrequent publication common in the early 20th century makes it hard to research the origins of terms. Authors would often develop an idea and share it with colleagues long before they published it (up to several decades). So the best way to source these claims is likely to by finding a good secondary source. Levy's book Basic Set Theory has detailed attributions of many of the basic concepts of set theory, and I would recommend that as a place to start. Also, remember that sometimes there is a standard, well known attribution of a theorem or idea to a particular person even though that person did not actually develop the theorem. I'm not sure what to do on WP in such cases. &mdash; Carl (CBM · talk) 15:27, 8 October 2007 (UTC)

I agree that if a secondary source is available, at least that could be cited. In the other case, where no secondary source has been located, probably the academically-rigorous approach would be (to use the example above):
 * The standard construction of the naturals, due usually [customarily] attributed to John von Neumann1, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:
 * s(a) = a ∪ { a }.


 * 1 Similar notions appear in X (reputable source here), Y (reputable source here), Z (reputable source here). [optional: The exact source is unclear.]

Bill Wvbailey 15:59, 8 October 2007 (UTC)

Merge with natural numbers?
After editing this page, I noticed that most of its content is duplicated at natural number. Do we really need a separate page?

Quux0r 07:24, 13 October 2007 (UTC)

Terrific prose but
To the editor: the writing under section Problem is great, an really should be preserved on a math blog before somebody comes along and deletes it for being "unencyclopedic" (as if formal math can ever be conceptualized in an encyclopedic manner, discounting prodigies). I especially love this insight: "After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands." Yes, editor, please save this somewhere. SamuelRiv (talk) 03:02, 18 March 2013 (UTC)

Is the purpose of this page to construct individual natural numbers, or, to construct the set of all natural numbers?
Some of the sections discuss only the first topic (construct individual natural numbers, for instance as an equivalence class under the equinumerosity relation, or as a set like in the von Neumann section). The second topic (the set of all natural numbers) is mentioned in the first section but not in the other ones. — Preceding unsigned comment added by MvH (talk • contribs) 15:02, 21 February 2020 (UTC)

Why are we doing this?
I realize this question doesn't always have an answer when it comes to mathematics, but in this case, it seems to me like there is some kind of motivation hinted at but never explicitly spelled out. The article on set theory itself states that "[s]et theory is commonly employed as a foundational system for mathematics", so I guess it's that? Just a short subclause in the introduction would be enough to make this more comprehensible to the layperson. --129.13.72.197 (talk) 17:53, 23 July 2020 (UTC)

Ackermann coding
Shouldn't the Ackermann coding be added to this article? It is a set-theoretic definition of natural numbers. BIT predicate is linked from this article which is good, I'm wondering could the history section of that article be copied over (I haven't edited on wikipedia in forever)? --Erinius (talk) 21:59, 25 February 2021 (UTC)

Erinius, this goes with the same thing that Quux0r pointed out. In the "Constructions based on set theory" section of the Natural Numbers page, the Zermelo ordinals are also cited as possible set-theoretic definition of the natural numbers. The fact that this and the Ackermann coding are both missing from this article is evidence that it has room for improvement. However, I am not qualified to make such an edit because I am still learning set theory (which is how I discovered this article), and this is the first time I have ever used Wikipedia, and I don't want to do something wrong. -- Shkotsi (talk) 04:46, 10 May 2021 (UTC)