Talk:Set theory/Draft

Nice work! Boris Tsirelson 18:22, 16 September 2009 (UTC)

Thanks! I only joined CZ because I thought the world needed a decent introductory account of set theory :-) Mark Wainwright 04:22, 9 May 2010 (UTC)

Are you sure the world needs nothing else? :-) Boris Tsirelson 07:57, 9 May 2010 (UTC)

It may well do, but perhaps nothing that I am so easily able to supply. Mark Wainwright 10:44, 20 May 2010 (UTC)

Probably this article is approvable. For now I am not ready to approve it, because some finer points are beyond my competence. I understand the given text about that, but I have no other sources to be sure. Namely:

• "An ingenious axiom of Goedel's, Limitation of Size";
• "Montague proved in 1961 that ZF cannot be finitely axiomatised";
• "NF ... is finitely axiomatisable";
• "NFU, whose consistency is implied by that of simple type theory".

Boris Tsirelson 08:19, 18 May 2010 (UTC)

That would be great. A web search suggests I was completely wrong about Limitation of Size and it was von Neumann. Also if http://planetmath.org/encyclopedia/Class.html is correct I shouldn't have claimed it as subsuming Powerset. I've amended the article accordingly. Thanks for the glitches below, which I see someone has fixed now. I can't provide references but I'll e-mail you if I can work out how. Mark Wainwright 10:44, 20 May 2010 (UTC)

Also I observe some minor errors:

• "difference X-Y or X\Y contains of all those" — either "consists of all" or "contains all", I guess;
• "X∪Y ={x|P and Q}, X∩Y={x|P or Q}" — swap them.

Boris Tsirelson 08:34, 18 May 2010 (UTC)

---

Being advised by Mark (the author) I have found a source for "NF ... is finitely axiomatisable" here: "Stratified comprehension is an axiom scheme, which can be replaced with finitely many of its instances (a result of Hailperin). Using the finite axiomatization removes the necessity of referring to types at all in the definition of this theory."

Hailperin, T. [1944] A set of axioms for logic. Journal of Symbolic Logic 9, pp. 1-19.

Boris Tsirelson 18:16, 20 May 2010 (UTC)

I also see on the same page (by Holmes) the phrase "NFU: New Foundations with urelements. This system is consistent, ..." which I fail to understand (surely because I do not work in logic). Consistent relative to what?? How is it related to the phrase "NFU, whose consistency is implied by that of simple type theory"? Boris Tsirelson 18:33, 20 May 2010 (UTC)

Yes, ZF cannot be finitely axiomatised:

Montague. Semantic closure and non-finite axiomatizability. In Infinitistic Methods, pages 45–69. Pergamon, 1961.

K. Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

Boris Tsirelson 18:46, 20 May 2010 (UTC)

Yes, consistency of NFU is implied by that of simple type theory:

R. Jensen. On the consistency of a slight(?) modification of Quine's New Foundations. Synthese, vol. 19 (1968/69), pp. 250/263.

M. Boffa, The consistency problem for NF. The Journal of Symbolic Logic, vol. 42, no. 2, 1977, pp. 215–220.

Boris Tsirelson 17:40, 22 May 2010 (UTC)

Thus, now I am ready to approve it. Boris Tsirelson 17:45, 22 May 2010 (UTC)

Power set and Cantor

I have replaced the statement:

"This was the basis of his demonstration that different sizes of infinity must exist."

It is historically incorrect. Cantor's first proof (1973) that the real numbers are uncountable did not use the power set idea. --Peter Schmitt 13:37, 25 May 2010 (UTC)

I see. (But "1973" means "1873", I guess.) Boris Tsirelson 13:52, 25 May 2010 (UTC)

Oops, of course.

G. Cantor: Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. (23.December 1873)

Journal für die reine und angewandte Mathematik. Band 77 (1874) 258-262 (pdf)

--Peter Schmitt 15:20, 25 May 2010 (UTC)

Thanks, and likewise for being much more scrupulous than I was about links and wiki formatting. I notice you've also made some minor stylistic changes, which I don't particularly prefer but don't propose to argue about except one which I have changed back faute de mieux: I think 'The difference X-Y' was more correct than 'The difference, written X-Y', since in the latter the noun phrase 'The difference' does not mention which difference. One could I suppose write 'The difference of X and Y, written ...' but the given form is conciser (and grammatically parallel to 'my cousin John'). Mark Wainwright 15:28, 27 May 2010 (UTC)

Then let us use the same style for all three operations. It less "concise", but it makes a difference between the operation and the notation. This is, I think, particularly important when there are two notations given. --Peter Schmitt 17:05, 27 May 2010 (UTC)

Notation

The bar may be the commonest notation, as stated in the article (though I don't know how you'd tell), but shouldn't the colon notation be mentioned too, as it's pretty common? And French mathematicians seem to use semicolons. Peter Jackson 14:48, 25 May 2010 (UTC)

Sure, but this page is a summary of set theory (with a personal touch) and to discuss all variations of notation does not fit into it. This can be done elsewhere, where more details are discussed. (There are variants for complement, as well, and for negation.) (Moreover, the variants are intuitive and easy to understand.) --Peter Schmitt 15:05, 25 May 2010 (UTC)

I took the same view as Peter; the aim is to present the ideas and just enough notation for the reader to understand the rest of the article. As to common, you're right, I didn't do a survey, I just stuck my finger in the air and thought I'd seen more bars. (I have no preference; indeed the history shows I originally put a colon in one place and corrected it for consistency later.) To avoid questions about the 'commonest' I've reworded to 'A common notation uses ...' which as a bonus is slightly conciser too. Mark Wainwright 15:28, 27 May 2010 (UTC)

Approval date

As far as I'm concerned, if an Editor on the 22nd, say, sets an Approval date for, say, the 25th, and then someone comes along in the meantime and makes a number of changes, as Peter has done here, the Approval date should then be delayed *another* three days, not just having a change make in the date of *when* the Editor approved things. The whole point of a delay between a proposed approval and the date of the approval is to give other people time to look at things. Or such is my understanding of things. If you disagree with this, please feel free to contact Matt Innes or Joe Quick about it. Thanks. Hayford Peirce 17:09, 26 May 2010 (UTC)

Let me first say that a few days earlier or later do not matter. So it is not a problem. And, of course, it is the purpose of the period before approval to allow some input and corrections and improvements. But I cannont follow your logic, Hayford: If, during this period, some changes are made that are considered an improvement and lead to an update of the version -- why should this delay approval? It is the result of the review period. (A delay may be useful if substantial changes happen. But in the case of only minor edits?) --Peter Schmitt 19:30, 26 May 2010 (UTC)

Hopefully, some day we'll have a charter that defines all that unambiguously, no one will violate it, and no one will demand more than that. Boris Tsirelson 19:46, 26 May 2010 (UTC)

This article is looking very good. I do suggest adding at least one reference before approval, if nothing more than an in-depth book that would cover the topic in greater detail. Fine job!Pat Palmer 21:44, 26 May 2010 (UTC)

I have Hausdorff's book purely on set theory, and a couple of graduate-level discrete mathematics texts with considerable set theory. How much citing is needed, as opposed to putting them in the bibliography? (The Lange review book is actually decent). --Howard C. Berkowitz 21:55, 26 May 2010 (UTC)

My answer to Peter is simply the comment by Pat. Any time changes are made, enough to have the original nominating Editor change the date of his Nomination, then, at least in MY opinion, more time ought to be given for the other Citizens to vet the changes. Doesn't that make sense? In times past I *think* that when we were waiting for an Approval to be carried out, a *couple* of times an Editor asked a *Constable* to go into the text and correct a typo or misspelling. Am I wrong about this? Could be. But my own feeling remains that before an article is Approved, every Citizen should have at least 3 days to review the changes. And, of course, I *do* agree with Boris, that this is something that eventually should be very, very clearly spelled out somewhere. Hayford Peirce 22:03, 26 May 2010 (UTC)

Pat's comment has nothing to do with the version update. I still believe that there is no reason that mere copy edits (corrections of typos, formatting, etc.) should delay the approval date. (The current CZ:Approval Process does not even mention a date change -- only the updating of "now", moreover, the minimum approval period is 24 hours. This may be objectionable, of course, but it is current.) --Peter Schmitt 22:35, 26 May 2010 (UTC)

I was using Pat's comment as a *general* example of what I mean when I say that I think there ought to be a little more time allowed in these cases. He wrote, "I do suggest adding at least one reference before approval" -- that's the sort of update I think every Citizen should have to the time to furnish. I frankly can't see why Boris is in such an absolute rush to get these articles approved within three days of his first nominating them without giving time to other people to look at them. Many of these articles have been around for *years* -- is another two or three days waiting for Approval going to sink the Citizendium project? Hayford Peirce 15:42, 27 May 2010 (UTC)

I see that I missed the presence of references on the Bibliography page. The 3 that were there were all mooshed up and I just didn't see them what with all the "nominated for approval" and other notices. So that is good. Incidentally, guys, I am a she. Just to be honest, not only do I feel there should be no hurry to approve any article, I don't even like the entire idea of approval, because I feel that people should always be free to improve any article and also, the knowledge available about many topics continues to evolve. So I wish there were not any such thing as Approval, as I think it tends to make things competitive--once someone is seeking approval, they tend to push for it, which is their right the way the rules now work. So I am not going to block any specific article based on my dislike of approvals, but I don't feel that hurrying is necessary either. The article should be made as good as possible if it is has to get frozen in place.Pat Palmer 04:29, 29 May 2010 (UTC)

Sorry, Pat, about the he/she thing -- I've also wondered about this for a long time about Sandy Harris. But regarding your comments about the Approval stuff, then you are basically against the whole point of CZ -- that we *do* have Approved articles that *cannot* be touched. Sure, work on the draft version, and eventually it will become Approved. But if we don't have blocked Approved articles, then what's our difference from WP -- except politeness, which, of course, *does* count for a lot? Hayford Peirce 05:00, 29 May 2010 (UTC)

Pat, I had to revert your edits to the Bibliography: It is only one book (with two editions cited).

As for Approvals: They are a defining feature of CZ. Their purpose is to "guarantee" that an article meets some quality requirements, is reliable and stable, and thus citable, not to stop improvements (or complete rewrites). I did not much care about approval because of several problems. Making even minor corrections was/is very difficult, and editing an acceptable article before nominating it for approval is very restricted (unless there are at least two active Editors who cooperate). Under these conditions I preferred to repair problems I happened to notice. But when Boris took the initiative things became easier, and I am happy to cooperate. (I do not mind a delay of a few days, though in some cases I wonder about the reasons given.) --Peter Schmitt 09:29, 29 May 2010 (UTC)

References

The book by Hausdorff is a classic, but it is not on set theory (in today's sense), but rather one of the first systematic books on topology. Many textbooks contain introductions on set theory, but why list any of them? The book by Halmos (a classic) is probably better recommandation than any of them.

As a modern survey, I shall probably add the Chapter in Gowers "Compendium", but I want to check first if it really fits.

As for references as footnotes: They are used rarely in mathematics (and will not help the reader). But I intend to move the sources mentioned by Boris to the Bibliography. This can happen later, because the subpages are not protected.

--Peter Schmitt 22:47, 26 May 2010 (UTC)