Talk:Naive set theory

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Old discussions (2002-2003) have been moved to Talk:Naive set theory/Archive1. The page was moved, so the old edit history is now with the archive page. I've copied back the two threads started this year. Happy editing, Wile E. Heresiarch 05:56, 6 Aug 2004 (UTC)

1 Origin of the term "naive"
2 "naive" vs. "naïve"
3 Set vs. Naive set theory
4 Venn diagrarms
5 Notes section
6 "naïve"
7 Formalist POV in this article
- 7.1 Set theory as used by mathematicians
8 semi-formal naive set theory, or, why it isn't difficult
9 various corrections
10 On being naive
11 Moderating propaganda against correct set theory
12 False impressions of Halmos
13 Outline of global solution
14 Semi-formal naive set theory
15 Usage data
16 Uniformedly at random reading
- 16.1 Devlin (1993)
17 Contradiction freedom's just another word ...

[edit] Origin of the term "naive"

"The name is perhaps derived from the title of Paul Halmos' book Naive Set Theory."

I'm not so sure about that. "Naive" is a general shop term used by mathematicians when talking about anything in general. The term means roughly "non-rigorous" (or at least not completely rigorous), informal heuristic arguments, or just general thinking out loud without worrying about justifying precisely every little detail. Thus, "naive" as opposed to "axiomatic" set theory. Revolver (25:05, October 24, 2003 (UTC))

I was surprised by this also. I had thought that Halmos's book was called Naïve Set Theory because it was about naïve set theory, not the other way around. -- Dominus 20:29, 9 May 2004 (UTC)

I could be wrong. I remember having a prof in grad school tell some students their approach to a problem was "naive", and them getting a bit offended, until he explained he explained it meant roughly the above. Revolver 06:39, 21 May 2005 (UTC)

[edit] "naive" vs. "naïve"

Is it really necessary to switch all the spellings to "naiive" (I don't even know how to get the double dots)? I know it's the original French, but I think it looks awkward to most eyes, it's listed as a "variant" spelling in every American and British dictionary I found, and it's harder to render. Revolver 02:30, 13 Mar 2004 (UTC)

Agreed. Widespread use of the term derives from Halmos, and his book was published in English as "Naive set theory". "Naïve set theory" is in use, but a Google search turns up many fewer hits than "naive set theory", and it doesn't help that WP floats to the top of the list. I'm inclined to revert to "naive". Wile E. Heresiarch 16:37, 4 Aug 2004 (UTC)

I have moved this article back to naive set theory and within the text reverted "naïve" to "naive" (except in a quotation). (1) "Naive set theory" is much more widely used on the web (about 4000 Google hits for "naive" compared to about 1000 for "naïve"). (2) Within WP, almost all links to this article are from "naive" instead of "naïve". (3) Halmos's influential book was "Naive set theory". (4) "Naive" is the ordinary English spelling. Happy editing, Wile E. Heresiarch 06:04, 6 Aug 2004 (UTC)

[edit] Set vs. Naive set theory

I think there is too much overlap between the articles Set and Naive set theory.

In reviewing the change history for Set, I find that the earliest versions of this article (can anyone tell me how to find the original version, the earliest I can find is as of 08:46, Sep 30, 2001) contained the following language prominently placed in the opening paragraph:

"For a discussion of the properties and axioms concerning the construction of sets, see Basic Set Theory and Set theory. Here we give only a brief overview of the concept." (The articles referred to have since been renamed as Naive set theory and Axiomatic set theory resp.)

As subsequent editors, added new information to the beginning of the article, the placement of this "brief overview" language, gradually moved further into the article, until now it is "buried" as the last sentence of the "Definitions of sets" section. Consequently I suspect that some new editors are unaware that some of the material being added to this article is already in, or should be added to Naive set theory or even Axiomatic set theory (e.g. Well foundedness? Hypersets?).

If it is agreed that, Set is supposed to be a "brief overview" of the idea of a set, while Naive set theory and Axiomatic set theory give more detail, I propose two things:

Add something like: "This article gives only a brief overview of sets, for a more detailed discussion see Naive set theory and Axiomatic set theory." to the opening section of the article Set.
Move much of what is in the article Set to Naive set theory or Axiomatic set theory.

Comments?

Paul August 20:23, Aug 16, 2004 (UTC)

I've now made all of the above proposed changes. Paul August 21:07, Aug 27, 2004 (UTC)

[edit] Venn diagrarms

I've removed the Venn diagrams in the "Unions, Intersections and relative compliments" section, because the third diagram was an XOR not a relative compliment. I'm not sure we really need diagrams here since they exists in the individual articles Union, Intersection and complement (set theory). (Unsigned comment by me from 20:54, Aug 17, 2004. Paul August ☎ 16:36, May 20, 2005 (UTC))

[edit] Notes section

What's with the text size? You would think this was a contract or an advertisement, not an encyclopedia!

Brianjd 05:22, Sep 25, 2004 (UTC)

I've changed the text in the section to the normal font size. Paul August 16:42, Oct 5, 2004 (UTC)

[edit] "naïve"

Since Wikipedia is an encyclopedia, wouldn't this article title's proper spelling be with an umlaut? Naïve with an umlaut is proper English, right? Is there a different rule regarding "naive set theory" over this? WhisperToMe 05:23, 27 Dec 2004 (UTC)

See the section "naive" vs. "naïve" above. Paul August ☎ 05:43, Dec 27, 2004 (UTC)

To my knowledge, both spellings are acceptable in English. Also, it's a diaresis, not an umlaut. Umlaut is only used for German words, where it signifies the eponymous vowel inflection called "umlaut". -℘yrop (talk) 06:04, Dec 27, 2004 (UTC)

User:WhisperToMe, I have responded on your talk page: in short, no. For the benefit of other readers here is a link: [1] Wile E. Heresiarch 07:01, 27 Dec 2004 (UTC)

Also of note, umlauts and diareses are only identical when typed, not when written. Umlauts are meant to resemble old German 'e's, while diareses are meant to... well, not. They are just dots. They may be used in the original French, but that's just a coincidence. "Proper" English requires them, too. This avoids turning the "ai" into a dipthong. Similarly, coöperate technically should have them. Obviously, we should leave them out in the title, but in when in quotes. To illustrate, what would you suggest would be the proper way to handle a book title with a misspelled word? Unless it was later corrected (Most preferable during the author's lifetime), it would be proper to leave it misspelled. This does not, of course, include changing the spelling in cases where, for example, the title was in Old or archaic English.

That's ridiculous. "Proper" English doesn't require it. I just checked several standard dictionaries, and cooperate with an umlaut (or whatever) wasn't even listed as an alternate spelling. You don't need these things to have dipthongs. Revolver 06:39, 21 May 2005 (UTC)

English words don’t use diareses (show me a single ö in the dictionary), and besides, cooperate really means co-operate. As far as I know, naïve is the proper spelling since it’s a non-English word, like the ` in vis-à-vis. The fact that English speakers use it doesn’t make it an English word.

As for relying on Google hits… in general, the internet is stupid. I’m sure there are words where genuine misspellings have more hits. —Frungi

[edit] Formalist POV in this article

I have a disagreement with the following passage:

Naive set theory is distinguished from axiomatic set theory by
the fact that the former regards sets as collections of objects,
called the elements or members of the set, whereas the latter
regards sets only as that which satisfies certain axioms.

Virtually all contemporary set theorists are in some sense working in "axiomatic set theory", in that they use the ZFC axioms, or some variant(s) thereof, as a reference point, and make reference to those axioms for various purposes. Very few, though, regard sets as "only that which satisfies certain axioms". The last would be an accurate description of, say, "group element" -- the notion of "group element" makes no sense by itself, and there is no "preferred" or "intended" group for a group element to be an element of, and a group is simply a structure that satisfies three or four simple axioms.

This is quite different from the situation in set theory. The ZFC axioms are by no means simple, and arbitrary structures satisfying them are not really of such great interest. What we're looking for, level-by-level at least, is the maximal structure satisfying the axioms at each rank, the intended universe of sets.

The above, of course, I write from a frankly realist POV. I haven't yet been able to come up with a good NPOV wording, but I'm pretty sure the existing wording isn't it. "Axiomatic" does not equal "formalist". --Trovatore 28 June 2005 07:48 (UTC)

Formalism duly suppressed (I think neutrally). Randall Holmes 22:37, 29 January 2006 (UTC)

It reads a little better than it did before, but I don't think it really does much to address the issue, which unfortunately is built into the very division of articles, and the "naive/axiomatic" dichotomy repeated in many places. In my ideal solution, axiomatic set theory would be merged into set theory, the latter would even-handedly discuss the conception whereby the axioms are motivated by the intended interpretation versus the reverse, and naive set theory would become a discussion of the subject matter of Halmos's book, with maybe a bit of discourse on naive comprehension and the intensional and extensional notions of set. See my recent additions to von Neumann universe for what I have in mind about an even-handed discussion of the two conceptions.

Here's the basic problem: It seems that the naive/axiomatic dichotomy is being used in two ways and thereby conflating them. "Naive set theory" is used to describe elementary methods, whereas modern research in set theory is brought under the "axiomatic" rubric. But then "axiomatic" is used to describe the way that formalists conceptualize what they're doing "in principle", though I think even they would admit it's not what they really do in practice (working set theorists have an interpretation in mind, whether or not they believe it corresponds to an underlying noumenon; you just can't make progress if you're not thinking of sets as objects).

It's a very difficult structural problem with the set theory articles as they exist; a little rewording here and there isn't going to solve it. I'd appreciate your thoughts on it. --Trovatore 00:40, 30 January 2006 (UTC)

I have said elsewhere that the dichotomy between naive and axiomatic set theory is false: there is just one set theory and it is usually ZFC. (the issue between ZFC and alternative set theories is a quite different issue). If you look carefully, that is what I am really saying in this article :-) The justification for many appeals to so-called "naive set theory" is "I can be careless about details here because this stuff doesn't really matter", and in mathematics this is never true. Ever. Randall Holmes 13:26, 30 January 2006 (UTC)

The "false dichotomy" is still built in to the division of material into articles in a seriously POV way; I don't think just saying somewhere that it's a false dichotomy is really going to help. I'm hoping to persuade people of this before I go making big changes, assuming I can figure out which ones to make. Can I take it that you agree with me on this project?

I'd quibble again on "one set theory and it's ZFC". Of course the way I'd put it is "...and it's the theory of the von Neumann universe". For example you seem to be using the term ZFC to cover, say ZFC+large cardinals. --Trovatore 14:42, 30 January 2006 (UTC)

What can I say but that you are right? The base theory of the von Neumann universe is probably less than ZFC; Zermelo + each set belongs to a rank which is a set + each ordinal is the index of a rank, and you are in business in the von Neumann universe... I'm not a formalist; I'm just as realist as you are (though my notions of what we are really talking about might be alarming :-), but I have the annoying habit of indexing domains of discourse by the theories which describe them. I'll try to reform, but I'm probably too old and set in my ways :-) Randall Holmes 15:16, 30 January 2006 (UTC)

So would I have your support on merging axiomatic set theory into set theory, and related changes outlined? Do you have any refinements of that plan to suggest, or any alternatives? --Trovatore 15:53, 30 January 2006 (UTC)

I have no objection having informal set theory articles alongside the article with the formal axiomatization. What I object to is claims that naive set theory responsibly practiced is different from what is formalizable in ZFC (or maybe Morse-Kelley) ("formalizable" as opposed to actually "formalized"), or even that it is especially difficult to avoid paradox. All one needs to do is restrict sets formed by comprehension to preexisting sets, for the most part (while allowing power set and union). I would rather like an intro article that informally introduced some axioms (say the axioms of Zermelo set theory) in the course of describing basic constructions; this article could point one to the advanced set theory article which begins with the full axiom list. Randall Holmes 17:26, 30 January 2006 (UTC)

I'm pretty sure I'd be opposed to merging axiomatic set theory into set theory. I think it's a far better to have set theory serving a pseudo-disambig page with links to the "naive" version and the rigorous version together with a discussion of the relationship between the two. Set theory is encounted by high school students and even grade school students. While I know Wikipedia is not supposed to be pedagogical it seems like too much a loss to start discussing formalism in the set theory article directly. At least discuss these issues with Paul August who put a lot of work into organizing the set theory articles as we have them now. -- Fropuff 16:51, 30 January 2006 (UTC)

I certainly agree that Paul should be involved (actually I gave him a heads-up about it a long time ago, though perhaps not with the specific merge idea.) You're probably right that there should be a highly visible fork to the sort of set theory that secondary-school students use. I just don't think the proper distinction is whether it's axiom-based or not; rather, it's an elementary-versus-advanced sort of distinction. Much of the current content of naive set theory might fit well in an article called elementary set theory. --Trovatore 17:16, 30 January 2006 (UTC)

Like Fropuff I'd probably be inclined to keep Set theory as an introduction to the various articles on set theory, although it could, and should, be expanded (in particular a history section).

For the record, when I arrived on the scene in July 2004, I found the structural organization of articles more or less as it exists today. The article "Set theory" looked like this pointing to this version of "Naive set theory" and this version of "Axiomatic set theory", as well as other articles. I also found an article called "Set" which looked like this. It seemed to me that there was too much overlap between the articles "Set" and "Naive set theory", see Talk:Set#Set vs. Naive set theory for my comments on the state of the set theory content as of Aug 2004. My primary "contribution" (such as it was) was to rewrite Set, moving much of its original content to "Naive set theory" and "Axiomatic set theory".

I'm familiar, to some extent, and sympathetic with, Travatore's concerns, see this discussion, I've been following the discussion above, and I'd be interested in figuring out if there is a better way to organize this content. I am willing help in any way possible. Perhaps it would be useful to agree on a set of articles we want to consider (e.g. Set Theory, Set, Naive set theory and Axiomatic set theory — others?), and then develope a proposal for how to better distribute this content across a proposed set of articles. That is, it might be better to approach this problem as globally as possible, as opposed to taking a more piecemeal approach, à la merging "Axiomatic set theory" into "Set theory", for example.

Paul August ☎ 18:20, 30 January 2006 (UTC)

Thanks, Paul. Let me break down to its bare bones why I don't see an alternative to the merge:

What we want to distinguish between is the set theory that set theorists do, versus the set theory of Venn diagrams and such that appears in elementary textbooks.
My objections to calling the former "axiomatic set theory" are known.
But there's no other good name for it, besides just plain "set theory". "Advanced set theory" and "research set theory" are unidiomatic and don't make natural links from other articles; "formal set theory" is more idiomatic but just as question-begging.
So barring some other good suggestion, I think set theory needs to be about what set theorists do, with a prominent link to elementary set theory (compare Boolean algebra, Boolean logic, though that's a little different because of the count-noun-v-mass-noun thing). --Trovatore 20:45, 30 January 2006 (UTC)

A side remark—"elementary set theory" could also be called "abstract set theory", as a categorist would have it (that is, sets are not individuals); Randall was alluding to this point, I think. --Trovatore 20:45, 30 January 2006 (UTC)

What do you propose we do with the content in Set and Naive set theory? And will there be no article called {{Axiomatic set theory]]? Paul August ☎ 21:12, 30 January 2006 (UTC)

This is a little off-the-cuff, but I think set and naive set theory could be merged into elementary set theory; I don't really know why we need separate articles. My preference would be to have axiomatic set theory a redirect to set theory, though I could also see it as a discussion of various alternative axiomatic treatments (ZFC, NF, anti-foundation axioms, maybe alternative set theory). --Trovatore 21:19, 30 January 2006 (UTC)

Trovatore draws a distinction between the set theory that set theorists do and the set theory that grade school students learn, but I think there is a very important category that is being overlooked here. That is, set theory as used by mathematicians that are not working directly on the foundations of mathematics. I think 95% of the time that mathematicians, physicists, etc. use set theory they don't need to worry about the rigorous formulation. I have certainly never worried much about it. I've (almost) never needed anything beyond what's in Halmos's book. To me naive set theory is the set theory that working mathematicians use as well as the set theory upon which set theorist base their intuition. For that reason, I would be opposed to renaming it elementary set theory. -- Fropuff 21:51, 30 January 2006 (UTC)

But what is in Halmos's book is a semi-formal presentation of the entire theory ZFC; it is the same subject exactly as axiomatic set theory. Randall Holmes 22:27, 30 January 2006 (UTC)

As far as merging set with naive set theory: we have a precedent for keeping the structure separate from the theory that studies it. Witness group vs group theory, graph vs. graph theory, and many others. -- Fropuff 21:51, 30 January 2006 (UTC)

Yes in my opinion we certainly need an article called "Set" which gives a "definition" of a set. It also seems appropriate to give a few basic ideas concerning sets in that article. Paul August ☎ 22:34, 30 January 2006 (UTC)

I have no real objection to a separate set article. As I said it was a fast first impression of how to answer your question about the overlap. Maybe a better answer is that the overlap just isn't so bad. --Trovatore 22:40, 30 January 2006 (UTC)

[edit] Set theory as used by mathematicians

This is in response to Fropuff's remarks a few paragraphs up:

...I think there is a very important category that is being overlooked here. That is, set theory as used by mathematicians that are not working directly on the foundations of mathematics. I think 95% of the time that mathematicians, physicists, etc. use set theory they don't need to worry about the rigorous formulation.

Well, neither do set theorists, really, as you say 95% of the time. But non-set-theorist mathematicians do use the results of foundational researches, and know more about them than they think they do (though less, of course, than they ought to :-). For example, they know how to apply Zorn's Lemma, and they know to expect that any set of reals they can actually define, will be Lebesgue measurable. So I think set theory as used by mathematicians fits just fine into what I foresee as the general set theory article, into which axiomatic set theory should be merged. Naturally it would be near the top of the article, whereas axiomatizations and advanced techniques would come later. --Trovatore 23:52, 30 January 2006 (UTC)

I guess as long as the set theory article had a prominent link to the naive set theory article and the set article is kept pretty basic, I don't having any major objections to merging axiomatic set theory in with set theory. Modern set theory is, after all, axiomatic; whether we choose to think about it or not. Perhaps we should continue the discussion at Talk:Set theory. -- Fropuff 02:03, 31 January 2006 (UTC)

[edit] semi-formal naive set theory, or, why it isn't difficult

Here is an example of a naive presentation which will do everything naive consumers probably want and which will not lead to paradox (all formalizable in ZFC).

As in the existing article, say that everything we do initially is carried on in a universe of discourse U. It is reasonable to stipulate that any set in U also has all its elements in U (we don't need to use the word transitive when we say this). (It might also be reasonable to stipulate that any subset of an element in U is also in U, but that is optional, and we certainly won't even whisper supertransitive).

Then say that for any property of elements of U, there is a subset of U consisting of everything with that property, which lives in the larger universe P(U)). If we stipulated that all subsets of elements of U are in U, then we would have $\{x \in A \mid P(x)\}$ in the universe U if A is in U, which is an advantage; otherwise subsets of elements of U get promoted to P(U). Notice that I said "property"; not a word about formulas of first-order logic!

We can say that the process of construction of power sets can be repeated any desired finite number of times (any property of elements of Pⁿ(U) determines an element of Pⁿ⁺¹(U)). The resulting world satisfies all the axioms of Zermelo set theory (as long as it is implicitly understood that everything is in some Pⁿ(U). Any instances of replacement that are not satisfied are the result of recursion on the universe-constructing machinery, which is not a naive application of set theory.

The specific recursion leading to the natural numbers can simply be presented and it can be asserted that such a set is provided. An easy way to do this is to close U under the construction of finite sets. This gives us each natural number in U and the set of natural numbers in P(U) (if we don't just provide it in U itself).

It's not that this is easy; some of these ideas are hard. But it is easy to avoid paradox: restrict all comprehensions to sets already formed or power sets of sets already formed (and allow construction of finite sets of previously constructed objects). No sophisticated logic needed. Randall Holmes 17:58, 30 January 2006 (UTC)

JA: Well, at least I know what that means for me — blue suit, no black tie — but what do the ladies wear? Jon Awbrey 18:36, 30 January 2006 (UTC)

RH: Please note that this is the meta-presentation, not the presentation... Randall Holmes 19:34, 30 January 2006 (UTC)

RH: Mostly for fun, the presentation. Ta da! Doubtless too convoluted. Randall Holmes 21:55, 30 January 2006 (UTC)

[edit] various corrections

I corrected the ascription of the paradoxes to Cantor's set theory (this is unclear, while Frege was definitely implicated). I remarked that naive set theory is not always the inconsistent set theory anyway: Halmos's book is about ZFC with all the axioms listed. Further, I removed the statement that axiomatic set theory is abstruse and has no bearing on mathematics; this simply isn't true (it's not that abstruse -- witness Halmos -- and it has some definite bearing). Randall Holmes 22:24, 29 January 2006 (UTC)

[edit] On being naive

JA: The meaning of the word "naive" in this context is synonymous with "native" and "natural", as in "native intuition" and the "natural light of reason" (Galileo's il lume naturale). It refers to that distributed but diverse intuitive conception of bunchiness that nature's evolution gave us at birth and that we nurture through comparative and critical reflection on the available panoply of provisional formalizations. The word was used this way in philosophy generally long before Halmos and others wrote the mathematical books on it. Though prior in time, the issue of (naive) set (folk) theory is perfectly analogous to the issue of the Church?Turing thesis about the eternally empirical and ever-intuitive notion of "what's a computation, anyway?" Thus, it is a pragmatic matter that cannot be resolved solely by considerations of deductive, exact, formal reasoning. The manifesto is ended ? Go in peace. Jon Awbrey 14:08, 30 January 2006 (UTC)

I don't really disagree with what you say here at all. Naive set theory is by definition not formalized (though the usual exact reference of this term is to a formal theory extracted from Frege by later workers; Extensionality + unrestricted Comprehension). But most naive set theory done nowadays is the sort which if formalized would end up in some relative of ZFC. Randall Holmes 15:30, 30 January 2006 (UTC)

Though I do have a specific opinion about this: I think that most "folk set theory" is either mereology or (folk) second-order logic (theory of universals). I don't believe that there is any folk set theory in the true sense of set theory, in which there are sets of sets (or if there is it is of very recent development), and this is one of the reasons that teaching set theory is hard. Randall Holmes 15:34, 30 January 2006 (UTC)

[edit] Moderating propaganda against correct set theory

To do set theory without risking paradox (in particular, to do set theory in a style which is formalizable in ZFC) is not especially difficult. Thus I eliminated the language which says it is a "much more difficult development". Perhaps the complaint is about logical technicalities about infinite axiom schemes? Of course, thinking about set theory is hard, but this is because it is quite abstract, and the difficulties about level of abstraction already exist in a "naive" treatment. Randall Holmes 15:27, 30 January 2006 (UTC)

[edit] False impressions of Halmos

Halmos's Naive set theory is actually a rigorous presentation of all the axioms of ZFC; it is axiomatic set theory. Every axiom, including the scheme of Replacement (which he calls Substitution) is presented quite precisely. Randall Holmes 22:35, 30 January 2006 (UTC)

I think this reflects a prejudice. One should remember that it is possible (though difficult) to present rigorously accurate mathematics without fancy symbolism... Halmos succeeds in doing this. Randall Holmes 22:38, 30 January 2006 (UTC)

Thanks for that report—I don't have Halmos's book available and have only barely glanced at it in the past. Part of the problem then seems to be his idiosyncratic naming. Set Theory from an Informal Perspective would have been a better name. The book may be a fine resource, but I think we don't need to ape his nomenclatural choices. --Trovatore 22:43, 30 January 2006 (UTC)

[edit] Outline of global solution

So I think Paul is right; it would be good to talk about a more global solution, and it occurs to me that maybe the way to go is to talk about the way I think things should end up rather than the deltas from the way they are now.

My vision would be that set theory should be the central article for the category, and would be from the mathematician/set theorist point of view. (I want to avoid separating these and giving the impression that what set theorists do is some backwater that's not relevant to other mathematicians. I know there are plenty of mathematicians that think that, but as a matter of respect for the subject, that idea shouldn't be built into the way the top set theory articles are structured.)

Content for other articles we've discussed:

Elementary set theory—a prominent fork, disambig'd at the top of the article. No need to include the mathematician view in this article; that goes in set theory itself. So this article is about set theory accessible to high-schoolers.
Naive set theory—this article should be more about the phrase than a particular theory, because it appears that it refers to various different things. Halmos's book should be mentioned, as should Frege's inconsistent theory (naive comprehension plus extensionality.) Could also mention abstract set theory as categorists see it.
Axiomatic set theory—redirect to set theory. I don't agree that modern set theory is "axiomatic whether we choose to think of it that way", but axiomatic treatments will be discussed in set theory, so it's an appropriate redir.
Set—this one doesn't matter that much to me. I suppose it'll inevitably wind up with some duplication, which I think is OK; let's just try to keep it within reasonable bounds and use links to other articles liberally.

Now for the main article, set theory, here's the outline I envision:

Lead section. Brief summary of article; talks about sets as collections of objects, mentions that they have many uses in mathematics and that mathematics can be coded in set theory, and that enough of set theory can be axiomatized to do most of mathematics. Remains neutral on whether the subject is defined by its axioms or by its intended interpretation. If the antinomies are mentioned, should not assert that axiomatization is the solution, but should mention that some consider them to have been solved by axiomatization, others by the cumulative hierarchy.
History. Try to keep reasonably brief; should have a {{seemain}} to history of set theory, which someone should write.
Applications. Talks about the unification of mathematics under set theory; how it made it possible to apply the results of one area of mathematics to problems from another.
Axiomatics. Mostly pointers to the various fine articles on specific axiomatic theories, with some description, explaining that ZFC is the most standard one, that many set theorits believe in extending ZFC with large cardinal axioms, discuss alternatives like NF and non-well-founded set theory.
Interpretations. The cumulative hierarchy (von Neumann universe) and anything else relevant. Explain the difference from Frege's logicist notion of sets as extensions of definable properties. added in a later edit 03:56, 31 January 2006 (UTC)
Branches of set theory. Subsections with {{seemain}}s and brief blurbs for infinitary combinatorics, descriptive set theory, inner model theory, large cardinals, determinacy, forcing, cardinal invariants, set-theoretic topology.
Open problems. Lots of choices here—I think the continuum hypothesis should be mentioned as a problem that some think is open, and some not.

This is a draft that I think would work well. If we can come to an agreement on something like it, then it would be reasonable to discuss logistics on how to get from here to there. Comments? --Trovatore 03:50, 31 January 2006 (UTC)

JA: Okay, this has gotten just plain silly. As one who got lambasted the other day for "rushing" to "reform" the whole of some area, simply because I used the word "co-ordinate" to refer to the interlinking of articles -- and anyone who knows me can tell you that I'm far too lazy for that sort of many-worlds-domination thing -- I feel fully justified in re-distributing a few from this large pile of stones.

JA: People who have any respect at all for the customary usages of the wider world tradition of set-theorists, not to mention the vernacular of mathemticians around the world, could hardly have a problem with the generally understood meaning of the phrase "naive set theory", which neither Halmos, one fine day, nor any other one individual "coined", but merely recognized in print what was long before them a widely established subject matter, under this or some other form of words. It's like the same way that some communities use "arithmetic" to mean what others call "elementary number theory", not "elementary" in the sense of "dummy's guide to", but simply treating numbers as primitive elements. So why is there this persistent need in Wikipedia to oversimplify the situation as it is? ZF is not all there is to axiomatic set theory. Axiomatic set theory is not all there is to set theory. Set theory is not all there is to mathematics. FOPC is not all there is to PC. PC is not all there is to logic. Logic is not all there is to mathematics. Yes, there are schools of thought here and there who have asserted and who will go on asserting any given form of reductionism that anyone can think up. And they are entitled by their POV. But anyone who thinks that these issues are foregone conclusions, or that there is some kind of widespread consensus on them, simply needs to get out more. Jon Awbrey 04:36, 31 January 2006 (UTC)

It's a little hard to figure out how your remarks relate to my draft. I don't think I'm suggesting any oversimplification or reductionism; I'm proposing to remove a categorization that, in my view, forced set theory into unnatural categories (while, I hope, leaving the articles acceptable to those who don't think those categories are unnatural). Yes, it's an ambitious outline, but that's because the problem is very difficult to fix incrementally; I've given it considerable thought over many months, and I was responding to Paul's suggestion to think about the problem globally. The invitation to all contributors stands; I'd like specific comments on both the thrust of the plan and its details. --Trovatore 16:39, 31 January 2006 (UTC)

There is nothing wrong with planning general organization of articles in a subject with due consultation. I particularly applaud the point that "naive set theory" refers to at least two very different things (which often seem to be conflated in the minds of those who use the term carelessly: on the one hand we have informal uses of set theory (which, as with work in any mathematical subject, should be formalizable on demand, and so should be compatible with ZFC -- unless one actually adopts some other set theory as one's standard) and on the other hand we have the specific (completely formalized!) inconsistent set theory best ascribed to Frege, which fell foul of the paradoxes. Randall Holmes 16:55, 31 January 2006 (UTC)

I like Trovatore's draft for the most part. My biggest objection is to the name elementary set theory. At some point someone is going to complain that such an article is a "pedagogical fork" of set theory and asked that it moved to wikibooks or merged in with set theory. Besides, elementary set theory may be insulting to someone who doesn't believe it's very elementary. I prefer the name naive set theory but unfortunately that term may mean more than one thing (an inconsistent set theory or consistent formalizable theory presented from a naive point of view). I suggest we use the title informal set theory instead. Which is to say that it is modern set theory presented from an informal point of view (i.e. without axioms). -- Fropuff 18:11, 31 January 2006 (UTC)

Thanks for the general vote of confidence. To be honest with you, I find the term "naive" more insulting than elementary. However you try to explain it as a term of art, "naive" is going to retain a sense of a normative preference for axiomatization; an implication will remain that non-axiomatic presentations cannot be "sophisticated". The point about elementary set theory being a pedagogical fork has some merit and I'm not wedded to the existence of such an article. But I feel strongly that ordinary-mathematician set theory should go in just-plain set theory, along with the set theory that set theorists do; it should not be shipped out to either "elementary" or "naive" --Trovatore 18:23, 31 January 2006 (UTC)

Somehow I missed your last sentence. Again, I think that should go in set theory. My central point is that the structure of the article should not imply that working set theorists are formalists; what set theorists mostly do is informal set theory. --Trovatore 18:26, 31 January 2006 (UTC)

I do really think we should have an informal article on set theory as understood by nonspecialists. To me the name informal set theory seems the most accurate and the least POV. I don't think having a separate article on such implies that what set-theorists do is always formal; no more than having a separate article on ZFC implies that set-theorists don't study ZFC. Rather the set theory article should have prominent links to both with a clear explanation of what set theorists really do. -- Fropuff 18:47, 31 January 2006 (UTC)

I don't quite understand the rationale for having a separate page for the theory as understood by nonspecialists. We don't have one of those for, say, group theory, even though lots of non-algebraists use a little group theory from time to time. Do you really feel that set theory as used by the bulk of mathematicians is different in kind from what set theorists do? If so, couldn't that viewpoint be presented as what some people believe, in the set theory article, rather than building it in to the article distribution and thus reifying it? Note that I propose that the set theory article will treat the informal view first, so there's no implication that mathematicians in general bother with the axiomatics of set theory. --Trovatore 18:54, 31 January 2006 (UTC)

I'm not a set-theorist so I make no claim to understanding what set-theorist actually do. All I'm trying to say is that I think we should have an article on basic (or elementary if you will) set theory presented from an informal point of view (i.e. without axioms). And I prefer the name informal set theory to elementary set theory. Just my 2 cents; take it or leave it. -- Fropuff 19:03, 31 January 2006 (UTC)

So I certainly agree we should have that content. Our only difference seems to be whether it should be a separate article. Could you live with that content being in the main set theory article, as opposed to a fork? Note again that my proposal has the informal content first. --Trovatore 19:05, 31 January 2006 (UTC)

You know, now that I think about it, I might be coming around to your point of view here. If we try to take all the content of the current naive set theory that really should be called "informal set theory", dump it into the set theory article ahead of both axiomatizations and branches of set theory, it means that branches of set theory will come way late in the article, maybe later than I'd like. So perhaps a blurb about informal set theory in the main article, with a {{seemain}} to informal set theory, and rigorous neutrality about its relative value to axiomatizations. (Then I think I'd like branches to come before axiomatizations, if that's OK with everyone.) --Trovatore 19:11, 31 January 2006 (UTC)

That sounds good to me. I was going to say that try to merge all the informal stuff into the set theory article that article is going to become rather long and hard to manage/navigate. The {{seemain}} is a good idea. -- Fropuff 19:20, 31 January 2006 (UTC)

Ok I think I can support this general program. I think the above idea's for the "Set theory" and "Informal set theory" articles are fine. I also think that Travatore's intended repurposing of the "Naive set theory" article is a good idea. I do have a question about the the term "axiomatic set theory", which Trovatore suggests redirecting to "Set theory". I have three books on my shelf: Takeuti and Zaring' Introduction to Axiomatic Set Theory, Quigley's Manual of Axiomatic Set Theory and Suppes Axiomatic Set Theory. My question is this, why did they bother to use the word "axiomatic"? Paul August ☎ 00:03, 1 February 2006 (UTC)

I don't know. I've never looked in any of those books. Maybe the authors are formalists? --Trovatore 00:12, 1 February 2006 (UTC)

Well perhaps, but I rather think they are trying to make a distinction between an older set theory which was not underpinned by a paradox-free axiomatic foundation, and modern set theory which is. For example, Patrick Suppes, Axiomatic Set Theory (1972), says In this book set theory is developed axiomatically rather than intuitively. He goes on to say that among several considerations for doing so, the most pressing is the discovery, made around 1900, of various paradoxes in naive, intuitive set theory, which admits the existence of sets of objects having any definite property whatsoever. Some particular restricted axiomatic approach is needed to avoid these paradoxes. Paul August ☎ 04:50, 1 February 2006 (UTC)

OK, so that's one point of view, with which I obviously don't agree (Suppes correctly identifies the problem, namely the identification of sets with extensions of definable properties, but then proposes the wrong solution). Still, it needs to be mentioned as a current point of view, which it certainly still is. My objection is to having it built in to the titles of articles and the distribution of material between them. That's why I don't see any role for the axiomatic set theory article for this purpose. --Trovatore 05:20, 1 February 2006 (UTC)

As I said above I'm agreeable to merging Axiomatic set theory into Set theory. But it is still unclear to me what to do about the term "axiomatic set theory". Just like the term "naive set theory", It is a term which has been widely used and needs to be discussed somewhere, even if it is a misnomer (or perhaps an anachronism?). Perhaps we should use the title "Axiomatic set theory" for that purpose, similar to what is intended for "Naive set theory". Paul August ☎ 17:23, 1 February 2006 (UTC)

Oh, I see. Yeah, I suppose the title could be used to discuss the viewpoint that it was axiomatics that saved set theory from Russell paradox. I wouldn't go so far as to call it an anachronism; it may even be the majority viewpoint. Even if it is, though, I still think the change needs to happen, because it's just very difficult to get across that it even is a viewpoint, if it's built in to the distribution of material among articles the way it is currently. --Trovatore 20:55, 1 February 2006 (UTC)

[edit] Semi-formal naive set theory

One might find the now more refined semi-formal naive set theory amusing. Randall Holmes 17:51, 31 January 2006 (UTC)

JA: Meantime, I found that black tie! $\triangleright\!\triangleleft$ Jon Awbrey 18:16, 31 January 2006 (UTC)

[edit] Usage data

Google exact phrase search data:
Basic set theory       -- 22,100 hits
Casual set theory      --      3 hits
Elementary set theory  -- 18,200 hits
Informal set theory    --    273 hits
Naive set theory       -- 32,100 hits
Naïve set theory       -- 34,700 hits
Prêt à Porter Set      --     28 hits
Semi-formal set theory --      0 hits

NB. Google is using fuzzy search even on exact phrases these days, so pretty much ignores fussiness over diaereses, except that it works disjunctively to bring in extra hits. Conclusion: I think we have a real opportunity to get in on the ground floor with that semiformal set demographic. Jon Awbrey 03:14, 1 February 2006 (UTC)

[edit] Uniformedly at random reading

[edit] Devlin (1993)

Devlin, K.J., The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd edition, Springer-Verlag, New York, NY, 1993.

Zermelo–Fraenkel set theory, which forms the main topic of the book, is a rigorous theory, based on a precise set of axioms. However, it is possible to develop the theory of sets considerably without any knowledge of those axioms. Indeed, the axioms can only be fully understood after the theory has been investigated to some extent. This state of affairs is to be expected. The concept of a 'set of objects' is a very intuitive one, and, with care, considerable, sound progress may be made on the basis of this intuition alone. Then, by analyzing the nature of the 'set' concept on the basis of that initial progress, the axioms may be 'discovered' in a perfectly natural manner.
Following standard practice, I refer to the initial, intuitive development as 'naive set theory'. A more descriptive, though less concise, title would be 'set theory from the naive viewpoint'. Once the axioms have been introduced, this account of 'naive set theory' can be re-read, without any changes being necessary, as the elementary development of axiomatic set theory. (Devlin, p. 1).

JA: Apparently the distinction betwen naive and otherwise is in the eye of the beholder, the outlook that develops in the exercise thereof. Jon Awbrey 03:48, 1 February 2006 (UTC)

I disagree utterly with the quote; I think that most people's "naive" constructions of the set notion are bound to be quite wrong! Most folk "set theory" is either mereology or second-order logic (both of these abstractions are part of folk knowledge to some extent); what is missing in either case is a proper understanding of sets as themselves elements of further sets. Moreover, I find that people who do have the ability to think correctly about the set notion in mathematics (after formal training) still don't necessarily see the difference between the mathematical notion of set and the informal notions when asked to explain in English... I do agree with the quote about the naive development ultimately being taken to be the opening of the axiomatic development (there is only one set theory) but set theory is not a particularly intuitive subject; intuition must be retrained! (my typo "restrained" wasn't bad, either). Randall Holmes 01:27, 11 February 2006 (UTC)

[edit] Contradiction freedom's just another word ...

JA: One of the things that we are supposed to have acquired an appreciation for in mathematics is the distinction between what we can prove and what we can merely imagine or wish to believe. In that regard, when it comes to horses and wishes about the consistency of any future axiomset that could stand up and be counted as a set theory, or claims about the equivalence of several different axiomsets for sets, well, about the most that I've ever seen any careful person claim for his or her pet system is that "it's free from the more obvious contradictions ..." (Kelley, more or less, I think). So let's try to keep that in mind. Jon Awbrey 05:06, 1 February 2006 (UTC)

Um—keep it in mind when doing what, exactly? --Trovatore 05:15, 1 February 2006 (UTC)

JA: When considering the project of "trying to make a distinction between an older set theory which was not underpinned by a paradox-free axiomatic foundation, and modern set theory which is" (Paul August ☎ 04:50, 1 February 2006 (UTC)). Jon Awbrey 05:26, 1 February 2006 (UTC)

Yes, that's a point, but a bit of a side issue, and I doubt Paul was claiming actual apodeictic knowledge of the consistency of any particular formal theory. --Trovatore 05:29, 1 February 2006 (UTC)

"Claims about the equivalence of different axiom sets for sets" can be stated and proved as formal results, and there is a cottage industry of set theorists who do this. Claims of the absolute consistency of particular sets of axioms can only rest on intuition; the bedrock on which my intuition stands is the cumulative hierarchy (Trovatore's von Neumann universe) and the formal theory whose consistency I believe to be justified by this intuition is the rather technical Zermelo + $Σ 2$ Replacement (weaker than ZF!). About ZF I think it is possible to have reasonable doubts: the difficulty is that intuitive reasoning about the cumulative hierarchy only justifies instances of Replacement which have a fair bit of absoluteness...(what I apply to get this level of Replacement is the Levy absoluteness lemma). This is equivalent, according to Solovay, to a rather nice subtheory of NFU! I think there is still a set of slides about this on my home page. Randall Holmes 20:06, 1 February 2006 (UTC)

Here is the actual link to my slides on the insufficiency of the intuition of the cumulative hierarchy to motivate ZFC. Randall Holmes 21:32, 1 February 2006 (UTC)

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