Talk:Set-theoretic definition of natural numbers

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[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 <<Axiom des Unendlichen>>). 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. — 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)

[edit] 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)