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)

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)

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)

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)