Talk:Set (mathematics)/Archive 1

Archive 1

Archive 2

Archive 3

Disambiguation

I started to do the disambiguation, but found that they are almost entirely for mathematical set. Would it not be better, and aid in accidental linking, if we just had "mathematical set" here and used a see also: for the other meanings?

Probably the mathematical meaning ought to be here, the other meanings being very much rarer in practice (as you have noticed). People will continue to link here by mistake anyway. Of couse, there's no harm in changing the links to set (mathematics) (or even the ugly mathematical set), since these are completely unambiguous and can be redirected wherever we want. --Zundark, Tuesday, April 9, 2002

If no one is going to fix the links, then I intend to move the page back again somewhen this week. --Zundark, Monday, April 15, 2002

I disambiguated all of the mythological links; I don't think that any of them are to the game. So somebody can either disambiguate the hundred or so remaining mathematical links, or Zundark can move the article back; I won't make that call. -- Toby Bartels 2002/04/17 —Preceding undated comment added 23:03, 17 April 2002 (UTC)

unheadered

I'm all for putting real text about a clearly "primary" meaning into an article that also points to less common meanings. I'm not sure whose comment it was, but someone argued that the disambiguating pointers should be at the top of the article in that case, and I'm inclined to agree. Perhaps I'll write up these suggestions more clearly in the disambiguation page. --LDC

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{x : x is a primary color}

This is not a very precise definition. Depending on whether you consider additive or subtractive color models , either yellow or green are primary colors. -- JeLuF 09:49 19 Jun 2003 (UTC)

You can change "a primary color" to "an additive primary color", if you like. But other additive colour models are also possible, so even this isn't completely precise. Perhaps we should replace it with a better example. --Zundark 10:19 19 Jun 2003 (UTC)

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Can someone put in a description of nested sets and representation of tree structures with this. I can't find a decent reference anywhere for this. -- Chris. — Preceding unsigned comment added by 212.19.85.137 (talk) 11:29, 9 March 2004 (UTC)

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:

1. 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.
2. Move much of what is in the article Set to Naive set theory or Axiomatic set theory.

Comments?

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

I have moved the sections on "Well-foundedness" and "Hypersets" to Axiomatic set theory, which I think is a more appropriate place for them - based on the idea expressed above that the Set article shold be a "brief overview". Paul August 07:34, Aug 18, 2004 (UTC)

I've made the the above proposed changes. Paul August 21:04, 27 August 2004 (UTC)

∪ symbol displays as box?

Someone changed each set union symbol "∪" (i.e &cup) to an uppercase U, because they were displaying as boxes. Is there a problem with rendering ∪? It looks ok for me (Safari, IE, OmniWeb on MAC OSX). Does anybody else have problems with this? Paul August 19:34, 31 August 2004 (UTC)

symbols displaying as boxes

I've got symbols &cup,&sub, &empty and &sube displaying as boxes (IE 6.0), and it looks very annoying. The reason is maybe that I use Russian as a default languge (Regional and Language Options settings), and have also got a set of Russian fonts installed, but my Windows version is English (non-localized). Everything looks fine in Opera, though. What character set does IE use to process these symbols?

(I've recently changed my default language to English and it still does not work for IE). Igor — Preceding unsigned comment added by 82.209.241.41 (talk) 08:50, 3 September 2004 (UTC)

Symbols and the set theory

I see too many symbols in set theory as little squares. I think to be the one that changed the "union" symbol to an upper U, but nothing can be done for other symbols. Referring to the article on TeX markup, I think the reason is that the article on Sets is not written using the latter language. I tested it, without saving, starting with <,math> (please ignore the ,) and ending with <\,math> and all the formulas included in between went ok with the usual symbols. Somebody should patiently change the source language. demaag.

You need to get the proper fonts so they show up. I'm not sure how you can do this, perhaps someone else can clarify. Or try a different browser (like Mozilla Firefox). Dysprosia 09:08, 5 September 2004 (UTC)

School curricula

I think it is an interesting remark to make that set theory at one point was included in school curricula. As I understand it, this was (in the West) mostly a reaction to the Sputnik shock (I suppose the Soviet bloc school system included set theory in its curriculum?). I don't really know how things are today, other than that at least some countries seem to have largely eliminated set theory from their curricula and don't introduce set theory notation until university. Prumpf 00:07, 11 Sep 2004 (UTC)

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User: 84.65.179.65 took exception to the sentence:

Basic set theory, having only been invented at the end of the 19th century, is now part of the elementary school curriculum.

With the comment: Took out 'part of the elementary school curriculum'. Where? In America? Didn't do it at my school. What's the relevance of this to the article anyway?!?)

I've tried to address these concerns by replacing the above sentence with:

Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school.

I don't know whether set theory is usually taught in elementary school now, or if it is where (maybe someone will inform us?). I do know that is was introduced, in many (if not most) parts of the United States, into the elementary school curriculum in the 1960s, as part of what was called "new math". The relevance of this to the article is that, although it is a relatively recent mathematical development, it is now (or was?) thought to be so fundamental as to warrant teaching it in elementary school. Of course many parts of the "new math" curriculum fell out of favor, although I think that set theory was one of the least criticized parts. Paul August 01:42, Sep 11, 2004 (UTC)

There's a joke that pertains here:

Progress in mathematics education:

1950

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?

1960

A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?

1970

A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100 and each element is worth $1.

(a) Make 100 dots representing the elements of the set M

(b) The set C representing costs of production contains 20 fewer points than set M. Represent the set C as a subset of the set M.

(c) What is the cardinality of the set P of profits?

1980

A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.

1990

By cutting down a forest full of beautiful trees, a logger makes $20.

(a) What do you think of this way of making money?

(b) How did the forest birds and squirrels feel?

(c) Draw a picture of the forest as you'd like it to look.

Paul August 01:56, Sep 11, 2004 (UTC)

Some version of this problem should also be politically correct, like 'His/her cost of production is $80 and his/her profit is $20' — Preceding unsigned comment added by Igor Kuchmienko (talk • contribs) 11:29, 11 September 2004 (UTC)

Symbols displayed as boxes

I confirm that under a French localized Windows XP/Internet Explorer 6.0, many symbols (like the U for union) are displayed as boxes. --Didier — Preceding unsigned comment added by 213.223.151.6 (talk) 09:52, 5 December 2004 (UTC)

Some reverted edits

Recently User:Kendrick Hang made some changes, some of which I have just reverted. The reversions were made, primarily to keep the article "a brief and basic introduction" and to reserve more detailed and complete treatments of these ideas for other articles (Naive set theory, Cardinality, Complement (set theory) etc.) as stated in the articles lead section and discussed above on this talk page. If anyone wants to discuss these changes further I'd be happy to do so. Paul August ☎ 17:57, 7 January 2005 (UTC)

I understand the need to keep the article to the basics, but wouldn't one assume that the basic set operations that most people read in an introductory discrete mathematics text are union, intersection, and difference? Maybe we could at least mention that relative complement is also known as a difference operation? If difference doesn't belong here, maybe we could put a link to where someone would be able to find more about it? -- Kendrick

Yes, union, intersection and difference are the most basic set operations. Although in my experience, the term "complement" is, by far, more commonly used than the term "difference". Following your suggestion, however, I've added that "relative complement" is also called "set theoretic difference". As for links, there is a link to Complement (set theory) which has a more detailed treatment of complements including a link to symmetric difference. Paul August ☎ 02:15, 9 January 2005 (UTC)

Disambig

i propose moving this to "set (mathematics)" and turning "set" into the disambig page, as there are currently 9 different entries linked to the disambig page. any serious objections? --Heah 17:19, 25 Apr 2005 (UTC)

I am not in principle opposed to doing the move, although I don't quite see the gain. The big question is, who is going to fix all the links, there are hundreds of them pointing to set. Oleg Alexandrov 17:29, 25 Apr 2005 (UTC)

There are just a bunch of articles with the name "set", and it would seem prudent and generally time saving to have that as the disambig page. Not a huge gain, but it's there. There certainly are a whole lot of pages that link here! Although looking at that list makes this whole thing less appealing, i'd be willing to fix the links, i guess, as i'm the one proposing the move. It'll take a lot of time but imo will be beneficial in the long run. --Heah 17:47, 25 Apr 2005 (UTC)

I think this is probably not a good idea. There are currently over 500 pages which link to Set, this is far more than the number of pages that link to all other entries for "set" on the disambiguation page, combined. The current disambiguation of set is an example of what is called "primary topic" disambiguation, which I think is the appropriate type of disambiguation in this case. Quoting from: Wikipedia:Disambiguation#Types of disambiguation:

"Primary topic" disambiguation: if one meaning is clearly predominant, it remains at "Mercury", the general title. The top of the article provides a link to the other meanings, or if there are a large number, to a page named "Mercury (disambiguation)". For example: the page Rome has a link at the top to a page named "Rome (disambiguation)" which lists other cities named Rome. The page Cream has a link to the page Cream (band) at the top.

Paul August ☎ 18:01, 25 April 2005 (UTC)

Removed inappropriate (in my view) text

I've removed the following text from the "introduction" section:

"The informality of this 'definition' of a set leaves clear that different sets are different; so the definition of a set goes hand in hand with a classification of its objects. Of course, sets share properties; but these properties are tightly connected with provisions in the definition of any given set. For example, we can't speak of combinatorics (see "cardinality" below) of uncountable sets, like the set of real numbers."

This text seems more like philosophy than mathematics, and frankly I don't really understand what exactly it is trying to say. In any case I think it is out of place here. The purpose of this article is to give "a brief and basic introduction" to sets.

Paul August ☎ 16:13, 29 April 2005 (UTC)

Living dragons

How do we know that set A is equal to the null set (where A is the set of living dragons)? — Preceding unsigned comment added by Taejo (talk • contribs) 09:14, 5 June 2005 (UTC)

Lead section

The lead section currently doesn't say what a set is, only that it is a concept in mathematics. Why not have the definition there? - Fredrik | talk 8 July 2005 07:25 (UTC)

I myself like it that way. The concept of set is a rather abstract one. I think it is good to have some rambling about its importance before getting down to business. But it was not me who wrote that, so let's see what others have to say. Oleg Alexandrov 8 July 2005 15:30 (UTC)

Oleg, you liike it which way? As it is, or as Fredrik suggests, with the content in the "definition" section moved to the lead? Paul August ☎ July 8, 2005 16:44 (UTC)

OK, I like it the way it is. :) Oleg Alexandrov 8 July 2005 17:28 (UTC)

I agree that the lead section could say something more about what a set is — but I think the content in the "definition" section (or something very much like it) should stay where it is. I will try to rework the lead and "definition" sections over the weekend. Paul August ☎ July 8, 2005 16:44 (UTC)

Well, the "definition" goes "Informally, ...". Fredrik | talk 8 July 2005 21:37 (UTC)

By the way I've often thought it might be nice if this article could be an FA. It would be nice to have a mathematics FA that was accessible to the general reader. What do you guys think? Paul August ☎ July 8, 2005 16:44 (UTC)

The thought occurred to me as well; this article seems quite accessible. Some more detailed history would be required, to elaborate on the importance assigned to sets in the intro paragraph. Fredrik | talk 8 July 2005 21:37 (UTC)

I'm not sure that this article, being a "brief and basic introduction" is the most appropriate article for much on the history of set theory. Paul August ☎ 18:57, July 11, 2005 (UTC)

OK I've had a go at expanding the lead and "definition" sections. Comments? Paul August ☎ 18:33, 11 July 2005 (UTC)

Thanks

This is a very clear page for beginners like myself, so I just wanted to thank everyone who's worked on it. Nice work folks. Lucidish 16:53, 15 August 2005 (UTC)

"see link"

I kind of dislike the style and usually wish to keep this kind of style to linking as minimal as possible as it is dissonant/disrupts reading style and doesn't flow too well in my opinion, but I wonder why it was used. Would one consider it acceptable to merely integrate it with the entire article? Ie. rather than discussing briefly about empty sets in one section, use something like

Main article: empty set

instead? — Preceding unsigned comment added by La goutte de pluie (talk • contribs) 21:17, 18 October 2005 (UTC)

SECTION: Cardinality of a set

This section takes the controversial stance that dragons do not exist, although there are many persons who believe that this isn't the case. Isn't it biased and a case of original research to include this personal opinion about the existence of dragons on an otherwise fine and upstanding page? 71.248.217.223 07:54, 12 November 2005 (UTC)

OK I've removed the reference to "living dagons". Paul August ☎ 11:33, 12 November 2005 (UTC)

I don't argue that the dragons don't exsit. One can prove that the set of living dragons is something we denote $\emptyset ,$ but don't worry too much about it as you are given choice to believe whether dragons exist or not, and anyway all this set and cardinality thing is an abstract math theory which would not influence the well-being of any more or less respectable dragon even for a moment. Oleg Alexandrov (talk) 17:43, 12 November 2005 (UTC)

We could simply use pink elephants or some other less controversial nonexistent creature. The objection is still silly though. Deco 21:46, 12 November 2005 (UTC)

Unordered?

Twice Fresheneesz, has added the qualifier "unordered" to the first sentence, which I've twice removed. I think that adding this is unnecessary, and can be misleading since ordered lists are also sets. Paul August ☎ 19:44, 21 May 2006 (UTC)

I suspect that the impetus behind Fresh's additions is the ongoing discussion he and I are having at talk:quadratic equation, where Fresh has a problem with my suggestion that the solution set of the quadratic equation be represented as an ordered set, which is denoted by the use of subscripts. However, I don't think the addition to this article was appropriate. -lethe ^{talk +} 19:48, 21 May 2006 (UTC)

My reversion of edit to "definition" section

I've reverted the recent edit of the "definition" section by User:Peak. My changes, and reasons for each, are the following:

• I reinserted the first sentence of the section: Like the concepts of point and line in Euclidian geometry, in mathematics, the terms "set" and "set membership" are fundamental objects used to define other mathematical objects, and so are not themselves formally defined. The purpose of this sentence is to make clear that while the section is titled "Definition", what follows is not strictly speaking a definition. It also explains the fundamental nature of sets.

• I changed the second sentence from: A set can be thought of as a well-defined collection of entities or objects. back to However, Informally, a set can be thought of as a well-defined collection of objects considered as a whole. I think this is better because:

• "informally" again helps make it clear, that what follows is not a formal definition.
• The qualification "considered as a whole" is important because it attempts to distinquish the set {1, 2, 3} from the three numbers 1, 2, and 3. For example, it is one thing not three things.
• I'm not sure adding the word "entities" is particulary useful.

• And I changed the third sentence from: The members of a set are called elements. back to: The objects of a set are called elements or members. This is better I think because here we are trying to "define" (again informally) the terms "element" and "member" (both frequently used mathematical synonyms) using the more primitive term "object", the term used in the pevious sentence.

Paul August ☎ 12:38, July 12, 2005 (UTC)

I think these changes are good, although really "a collection" is "one thing". If possible I'd prefer some wording that makes it clearer that, for example, the set containing the empty set is different from the empty set. A good analogy is a bag or box with things in it.

I'd also probably say "objects in or composing a set", to be more specific. Deco 22:16, 13 July 2005 (UTC)

"A set is a collection of objects considered as a whole." I think we should somewhere mention in the text that this is in fact saying a set is a set, and we can't get around that, because the concept of set is so basic a thing. The sentence still gives a good intuition about it. 85.156.185.105 10:46, 22 August 2006 (UTC)

Improper Subsets

Can we get a clarification on Improper Subsets? —The preceding unsigned comment was added by MLeg11 (talk • contribs) 15:03, 10 September 2006 (UTC)

That's not really a term anyone uses much. If you say "A is a subset of B", then A might or might not be equal to B. If you say "A is a proper subset of B", then A is definitely not equal to B. That's the only difference between "subset" and "proper subset". There's simply no need for a term "improper subset", and it isn't used. --Trovatore 19:06, 10 September 2006 (UTC)

Oh, I might also mention—because this is the sort of point on which people sometimes get confused—that if I say "A is a subset of B", I am not asserting that I don't know whether A is equal to B. I may know full well, and simply not be saying. This is not usually because I want to be difficult. More commonly, it's obvious from context whether or not A equals B, and there's just no need for me to repeat information that's clear to both of us. --Trovatore 20:10, 10 September 2006 (UTC)

Contradiction / clarification?

Under the "Definition" header, the article reads :

A set, unlike a multiset, cannot contain two or more identical elements.

However, under "Description", the article reads :

Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list, so {6, 11} = {11, 6} = {11, 11, 6, 11}.

The use of {11, 11, 6, 11} may be confusing to the casual reader, who has just been told that a set never has identical elements. Perhaps someone who writes better math-prose than I can edit the article to clarify why this notation is valid?

Best, -- Docether 15:55, 22 March 2007 (UTC)

It is not valid. I see a contradiction here as well and deleted the contradicting part (the repetitions). Thanks a lot for pointing at this! — Ocolon 17:44, 22 March 2007 (UTC)

There is no contradiction; I added a sentence to the article already to try to explain what is going on. Although the set itself "can only contain each element once", the set builder notation can list it as many times as desired. So once an element is listed once as an element, you can ignore it if it is listed again. For example, the set

{ pq in N : p is even and q is either 1 or a prime}

only includes the number 4 once, not twice, despite the fact that there are two different ways to write 4 in the form pq specified. CMummert · talk 18:46, 22 March 2007 (UTC)

Okay. Thank you for the lesson. :-) — Ocolon 18:50, 22 March 2007 (UTC)

Explanation of Symbols

Can someone tell me what the symbol in the third linebelow is ? It looks like an equals sign on the page ?

"Some basic properties of unions are:

A U B = B U A

A ⊆ A U B "

Where would I go to get a summary of the meaning of the set symbols ? Diggers2004 05:50, 11 April 2007 (UTC)

Hi, Diggers. The symbol you're complaining about is actually coded as "&sube;", which is an HTML entity reference to Unicode character 0x'2286'. It really ought to display like this:

$\mathrm {A} \subseteq \mathrm {A} \cup \mathrm {B} \,$

I think this article summarizes the symbols fairly well. You probably have a problem with the fonts installed in your system, and they can't represent all the Unicode symbols adequately. Anyway, besides the union and intersection symbols, the main relationships among sets are denoted with rounded off versions of <, ≤, >, and ≥. (Hopefully those symbols – less than, less or equal, greater than, greater or equal – display properly in your browser.) So we have

$\subset$ "is a subset of"; $\subseteq$ "is a subset, or equal to"; $\supset$ "is a superset of"; and $\supseteq$ "is a superset, or equal to".

I hope that's clear enough, and that you can see the right symbols now! DavidCBryant 12:15, 11 April 2007 (UTC)

Using double-braces

1={1} ?

{{1}}={1} ? 79.113.82.169 18:33, 1 August 2007 (UTC)

Why cant I write that? {1} = {{1}} ?

You need to nowiki it: <nowiki>{1} = {{1}}</nowiki>. (But it's mathematically incorrect anyway.) --Zundark 18:50, 1 August 2007 (UTC)

Ensemble

As far as I understand, set is sometimes referred to as ensemble, so this explanation could be included, too. 80.235.68.14 08:50, 30 August 2007 (UTC)

How many elements?

I recently made some minor edits to the section on cardinality, for instance prepending the modifier "Colloquially" to the very first sentence. User:Trovatore disagreed and undid them. The point I wished to make—or at least to acknowledge—is that talking about "how many" members there are is dicey when we move beyond the realm of finite sets. I'm sure we could engage in lengthy philosophical debate about the extent to which cardinality of finite sets should be seen as merely a special case. But I think that would be off the point. My view is that Wikipedia is not, nor should it be, a rigorous mathematical text. Rather, it should provide clear (and never incorrect or even misleading) information. We also should recall that we mathematicians do not hold the deed on phrases like "how many." When we are speaking among ourselves it is perfectly appropriate to restrict our usage to the agreed technical senses of terms. But in a general-audience encyclopedia like Wikipedia, we should meet our readers where they are, rather than expect them to ascend the ivory towers were we ourselves are so comfortable.

In everyday English counting means "ascertaining how many." The two are semantically identical in the real world. But we ourselves (thanks to Cantor) describe R as uncountable. So in our technical sense, one cannot count the reals. And this aligns nicely with the civilian notion of counting: I think the man in the street would feel more comfortable trying to count the members of N strung out on a number line—even though he'd know he could never finish the task—than he would trying to count the points on the real line. (I know perfectly well that the rationals are dense and nevertheless "countable," but that gets us back into our technical world of mathematical rigor, and my point is about explaining to laymen.) That's why I'd done those few edits. I wished to signal to Wikipedia's general readership that cardinality is not identical to, but essentially a generalization of, the everyday concept of "how many." Heck, maybe explicitly saying that is a better way of achieving my aim.

Anybody think I'm crazy? Anybody think my edits should be restored?—PaulTanenbaum 18:29, 1 September 2007 (UTC)

Well, I disagree with you that it's "dicey". I think the interpretation of cardinality as the answer to the question "how many elements" is exactly correct, not just a generalization but the completely canonical right generalization. Now, I acknowledge, there are those who disagree, though I think those who think it's in some sense the wrong generalization are a distinct minority. It would be good to work their views in somehow, but not by compromising the simple statement with weasel words. Instead, the ideal approach would be to research the dissident views and add a section about them (while leaving the initial explanation a direct statement in line with the way mathematicians and especially set theorists usually talk). --Trovatore 01:05, 2 September 2007 (UTC)

I'm flattered that you characterize my reluctance about absolutism as weasely. But anyway... You think it is the right generalization. Well, so do I. But, as I wrote above, philosophical debates miss my point. Even your very good suggestion about including a section on any minority opinions doesn't quite get it. Wikipedia articles about mathematical topics are not articles in a mathematical encyclopedia. I think we should explicitly accommodate the divergence between the colloquial and technical meanings of some of the terms. For instance, to the vast majority of Wikipedia users, counting and ascertaining how many are the same concept. This isn't a problem, but an opportunity... to share a beautiful generalization; our usage is a generalization. And all I'm arguing is that we shouldn't ignore or deny the typical readers' view of matters, but address it—at least in passing—and help them expand it.—PaulTanenbaum 15:04, 2 September 2007 (UTC)

The colloquial and technical meanings here precisely coincide; there simply is no divergence. When we say there are more real numbers than natural numbers, we are speaking the precise truth, and "more" means exactly what it means in natural language (as you said, it's the same as counting, and it's also the same as counting for us; we just need more objects to count with than there are natural numbers). Now, discovering this fact is nontrivial, but that's the more technical point that should be deferred to the deeper discussion. There is no need to compromise in the initial statement. --Trovatore 00:18, 3 September 2007 (UTC)

But then what can we mean when we say that the reals are UNcountable? No, Trovatore, I'm afraid that counting something, even for mathematicians, no, especially for mathematicians, means injectively mapping it to measly old Z$^{+}$. So when we ascertain the cardinality of a set like C or 2^R, you may choose to describe it as figuring out "how many," but whatever it is we've accomplished, it isn't a count of anything.—PaulTanenbaum 03:25, 3 September 2007 (UTC)

You count them with ordinals, of course. It is a count; you just have to go past a limit ordinal to get there. Don't take "uncountable" too literally; it's just a word. "How many" on the other hand is not just a phrase -- it is exactly and literally what cardinality is. --Trovatore 04:14, 3 September 2007 (UTC)

Of course I'm taking uncountable literally. I'm a mathematician, I pay careful attention to definitions. To count the members of a set means to map them bijectively to either [n] for some positive integer n, or to Z$^{+}$, as per, say, Van Nostrand's Mathematics Dictionary (sorry the rest of my library is at work). Or see the article right here on counting, which describes it as "starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers starting from two."

You write that "how many" is, in fact "exactly and literally what cardinality is." I note in passing that your statement is factually wrong: the literal definition of cardinality (according to Von Neumann, anyway) has to do with the least ordinal bijectively mappable. THAT is "literally what cardinality is," and the phrase "how many" appears nowhere in it. But, more importantly, you still appear to be acting as though mathematicians owned the English language. The argument you present is a technical one aimed for another mathematician; and I repeat that Wikipedia is not a scholarly mathematical publication. Wikipedia's guide to writing better articles advises: "make your article accessible and understandable for as many readers as possible." And that's exactly what the author(s) of cardinal number did: "In informal use, a cardinal number is what is normally referred to as a counting number," [my, what weasel words!] and "when dealing with infinite sets ... the size aspect is generalized by the cardinal numbers."—PaulTanenbaum 19:30, 3 September 2007 (UTC)

No, "uncountable" does not mean "you can't count it". There isn't any uniform precise meaning of "count", so it can't mean that (because "uncountable" does have a uniform precise meaning).

It's not about whether mathematicians "own" the English language. We are using the meaning of "how many" that the English language gives to that phrase. This fact is not obvious to non-mathematicians, but it's true nevertheless. Once non-mathematicians understand the issues, they will agree that cardinality measures how many elements there are in a set. --Trovatore 05:48, 4 September 2007 (UTC)

list v. set

A suggested change:

By contrast, a collection of elements in which multiplicity but not order is relevant is called a multiset. A collection of elements in which multiplicity and order are relevant is called a list. Other related concepts are described below.

I believe that was the definition in my linear algebra textbook last semester. Goodralph 02:18, 21 Jul 2004 (UTC)

I'd further second this reference if list was the only term for such a construct, but alas. . . --Liempt 16:57, 19 September 2007 (UTC)

if vs iff in mathematical definitions.

In mathematics, the use of "if" is in definitions is the common practice, and it is perfectly "precise". Quoting from Talk:iff:

Regarding "if/iff" convention for defs:

I've reinserted the comment about "if" being used conventionally in math defs. I'm sorry, I've read a lot of math books, and this is a common convention. Many definitions use the terminology "if", in the sense of "If P(X), then X is called blah" or "X is said to be blah if P(X)", yet not every definition uses "iff", and all definitions are intended to be "iff", because that's what definitions are. (To counter your remark, definitions are not intended to assert equivalencies; an equivalence is usually meant to indicate a statement saying two things imply each other that has to be PROVED...definitions aren't proved, they're declared, so it doesn't make sense to say e.g. "'R is an integral domain' is equivalent to 'R is a commutative ring with identity'" because these statements aren't "equivalent" in the ordinary sense of the term, one does not PROVE they're equivalent, that simply IS the definition of an integral domain. Here are several cases where the "if" convention is used in the wikipedia itself...

• "A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n" (from prime number)

• "If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b." (from integral domain...notice, the first use of the word is in the sense of a definition, hence only "if" is used (although "iff" would be correct as well), but the second IS an actual theorem (result) because the equivalent condition requires proof. So, for the second statement, the meaning would change if "iff" were replaced by "if", although for the first statement it doesn't matter.

• "In complex analysis, a function is called entire if it is defined on the whole complex plane and is holomorphic everywhere" (from entire function).

The list could go on. Revolver

I would like to point out that while a biconditional statement does not imply an equivalence, it is an equivalence relation. I'd also like to point out that even though this is used within wikipedia itself or within some math books (although not any ones above undergrad level that I've read), it does not make it a good idea. While it may save effort and save you from typing three words, it's going to give the uninformed the idea that these definitions are specifically not biconditionals, but ordinary conditionals, and by extension, not an equivalence relation or that there exists some example where the antecedent of the conditional holds true but not the precedent. --Liempt 16:57, 19 September 2007 (UTC)

Revolver is correct. "If" is used everywhere in mathematical definitions, both in Wikipedia and elsewhere. "iff" or "if and only if" is commonly reserved for use in biconditionals, which definitions are not. The use of "iff" or ""if and only if" here is particularly inappropriate as this is intended as a basic introductory and elementary level article, which could be read even by a grade school student. Paul August 14:43, Nov 9, 2004 (UTC)

I would say, in biconditionals and in theorems (although formally a theorem is a biconditional; but you are talking about grade school students here...). Mikkalai 18:45, 7 Jan 2005 (UTC)

I strongly disagree. You'll never see a formal definition use the ordinary conditional. The very nature of a definition is that the both the precedent of the conditional implies the antecedent and vice versa. If we use an ordinary conditional then there is nothing to imply, given A -> P(A), that P(A) -> A holds. Even Revolver said that it is correct to use the logical biconditional as opposed to the conditional. I am opposed to sending false messages to the general public out of pure sloth, and by using an ordinary condition, we're doing just that.--Liempt 16:57, 19 September 2007 (UTC)

More formal definitions of Union, Intersection, Complement, Subset

The following definitions were removed after I added them because they were "inappropriate" to the article:

• Subset - $\forall x\forall Y\forall Z((x\in Y\to x\in Z)\iff Y\subseteq Z)$
• Union - $\forall x\forall Y\forall Z((x\in Y\lor x\in Z)\iff x\in Y\cup Z)$
• Intersection - $\forall x\forall Y\forall Z((x\in Y\land x\in Z)\iff x\in Y\cap Z)$
• Complement - $\forall x\forall Y\forall Z((x\in Y\land x\notin Z)\iff x\in Y-Z)$

I'm not upset or anything of that nature, I'm just wondering why they were thought to be inappropriate. I think they are easily within the realm of an definitive article of "sets" which is what this aims to be. --Liempt 17:04, 19 September 2007 (UTC)

As stated at its top, This article gives a brief and basic introduction to what mathematicians call "intuitive" or "naive" set theory. I think most casual readers (e.g. the average high school student) will not be able to comprehend these formulas. So I think they will only be in the way here. In general we try to avoid using symbols when words suffice, quoting from our Manual of Style for mathematics: "Careful thought should be given to each formula included, and words should be used instead if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the ∀, ∃, and ∈ symbols." A more appropriate place for these formulas might be our articles subset, union (set theory), intersection (set theory), and complement (set theory). Paul August ☎ 17:53, 19 September 2007 (UTC)

That does make sense, and I do agree that the average high school student wont be able to understand these formulas but a mathematical object can only be rigorously and formally defined by a string of logical symbols. I did also try and summarize each of the formulas afterwards in plain English. Plus, high school students aren't the only people who read this article. Lots of undergraduate and graduate students alike use Wikipedia as a reference, and they may be able to understand these formula. Perhaps I should put them in naive set theory, but that's more or less a carbon copy of this article? I don't really think that they belong in axiomatic set theory. I suppose if we concede that this article is intended to be basic, I agree with you, but if we don't want it to be basic by intention, I don't. Also, while we're on the topic of appropriateness, do you think this article could use a section on the cartesian product? Anxiously awaiting your response, --Liempt 18:27, 19 September 2007 (UTC)

This article and our article "Naive set theory" have existed in parallel for a long time, and discussions about their overlap and relationship and how they should fit into the suite of set theory articles have been several (e.g. see: Talk:Naive set theory#Set vs. Naive set theory, User talk:Paul August/Archive4#systematic error in set theory pages Talk:Naive set theory#Formalist POV in this article, Talk:Naive set theory#Outline of global solution). In any case, I think the intent for this article is for it to be a very basic introduction to the content in Naive set theory, so I would leave Cartesian product for that article. Paul August ☎ 20:19, 19 September 2007 (UTC)

All right, I'll work on adding some more formal definitions to the articles for union, intersection, etc. . . and work on the naive set theory article. This was a productive conversation, it really gave me an idea of the intentions for this article, thanks. My only issue is that it'll be hard to get this article to feature article status (as per our collaboration goal) without going into very much in-depth mathematical detail. --Liempt 20:39, 19 September 2007 (UTC)

Anyway, the above are not strictly speaking definitions, but propositions, something that can be true or false (of course, these ones are true). When you make a definition you don't quantify out the free variables going into the relation being defined, and you don't connect the symbol being defined with its definition with $\iff$. Of course this point is a bit of a quibble, but from time to time you do see people conflating definitions with propositions, and I think it's worth making the point. --Trovatore 18:04, 19 September 2007 (UTC)

Fair enough. Glad you made said point. :D --Liempt 18:27, 19 September 2007 (UTC)

How many empty sets?

(I've copied the following discussion form my talk page)

I do not object to your desire to treat the empty set as unique. But I believe that it was I who had changed the set article to refer to "an empty set." The reason I made the change was to accommodate the context, which said (and now once again says) "A set can have zero members. Such a set is called the empty set." You must agree that this is an unhappy collision of indefinite and definite articles, of assertions of existence and (subtle) assertions of uniqueness. Furthermore, the expression "Such a set..." means in mathematical prose "any such set" or even "every such set." One way that this problem could be resolved is to rework the passage to something more like "There is a set with zero members, which is called the empty set." What say I just do that?

I'd also point out that both your change to the article and your accompanying edit summary—"only one empty set"—gloss over a legitimate contrary view: for some purposes, like strongly typed reasoning, it is desirable to distinguish, say, between the set of Beatles obtained by deleting Ringo from the set {Ringo} and the set of integers whose square is 2, because sets of Beatles are not the same as sets of integers. Yes, of course, one might posit some isomorphism between Beatle sets and various integer sets, and since that's an equivalence relation, those two sets are "the same" to within isomorphism. Hence my first sentence in previous paragraph.

If you wish to reply, please hit my talk page.—PaulTanenbaum 00:39, 17 September 2007 (UTC)

Hi Paul. I'm fine with your proposed change above. I'm aware of other views about the empty set but I don't think we need to address them in the "set" article. As an aside though, I can't help thinking that every Beatle who is in the set {Ringo} / {Ringo} is also an integer whose square is 2, and vice versa. Paul August ☎ 18:52, 17 September 2007 (UTC)

No, that certainly is not a rat hole we need to go down in the "set" article. And I quite like your observation that no such Beatle fails to be such an integer. When my connection becomes more reliable, I'll make the change we've agreed on. Regards—PaulTanenbaum 03:53, 18 September 2007 (UTC)

(end of copied text)

Paul August ☎ 18:11, 19 September 2007 (UTC)

But hold, there is no Beatle in that set; it's empty. I also would like to rephrase the above statement. A set can have zero cardinality but it shouldn't have zero members: I'd rather say it has no members. --Liempt 18:31, 19 September 2007 (UTC)

You are right, Liempt, that there is no Beatle in that set. But that in no way contradicts Paul August's observation. Every Beatle who is in that set is, in fact, such an integer. As to your second point, the peculiar Beatle set under discussion has zero members every bit as much as the set of all Beatles has four members.—PaulTanenbaum 02:10, 20 September 2007 (UTC)

I wasn't trying to contradict his observation, just pointing out something so it isn't so misleading (although, that's part of the pseudo-joke). The way he phrased it, it might be interpreted by a less-than-informed person that Ringo is an integer who's square is two. Just thought I'd point out that it's not ringo that's equal, it's the empty set. As for the whole zero as opposed to other phrasing thing, it's more of an issue to me of clunky sounding prose. You don't say "I have zero friends", you would probably say "I have no friends" or "I don't have any friends". However, I don't have any problem when it's a value being to refered to, like "The cardinality is zero.", but in the context of, "This set has zero members" it sounds wrong to me. Maybe I'm wrong; I probably am. Anyhoo, cheers. --Liempt 03:10, 20 September 2007 (UTC)

Cartesian Product

I have added a section on Cartesian product to the basic operations header. My justification for doing this is as follows:

• It's a very basic concept, requiring only knowledge of an ordered pair.
• It is almost always learned immediately after the union, intersection and complements in just about every textbook I've ever read (trust me, that's a lot of 'em).
• It provides a neat contrast to union's "addition", providing the article with the corresponding "multiplication".
• It allows us to neatly define the concept of a relation under the applications heading. This extremely important idea may provide your average high school student a relate-able reason as to study set theory that may not be immediately apparent within the rest of the article.

If one further objects, I welcome the opposition. Please don't hesitate to tell me why I'm wrong. Anywho, cheers! Liempt 07:08, 23 September 2007 (UTC)

No mention of extensionality?

Not sure what CBM means in the ToDo list. The section set#describing sets briefly describes both intensional and extensional approaches to specifying membership. What's lacking, a more expanded philosophical discussion of extensionality? PaulTanenbaum 01:50, 21 September 2007 (UTC)

Well, sort of, yeah. Not speaking for Carl here. But the distinction between the two concepts of set or class (which goes well beyond different notations for naming them) might be worth treating. Conflating the two is what got Frege into trouble and led to the Russell paradox. And it also speaks to the "different empty sets" issue you mentioned earlier. --Trovatore 02:00, 21 September 2007 (UTC)

I agree it goes beyond and could well deserve treating, and that it gets at the typiness of sets, too. As it happens, my question (which only sought clarification of Carl's intent) is a bit out of phase anyway, since, having just checked the archive of reviews, I see that all the comments date from January 07 (before I'd added any mention of intension/extension). Which brings up another question: why did the ToDo list suddenly appear? Is it a side-effect of the parent article's selection as CoTM?—192.12.67.10 02:11, 21 September 2007 (UTC)

Son of a gun! I'd timed out when I signed above. For completeness, 192.12.67.10 was me.—PaulTanenbaum 02:13, 21 September 2007 (UTC)

I put this article through peer review a while back, and added the results to the to-do list. CloudNine 07:03, 21 September 2007 (UTC)

I think it's safe to cross this one off the list, at least in the (assumed) context he added it. There's mention of both ways of declaring members of sets, extension and intension. There's a rephrasing of the axiom of extensionality within the article, and I don't think that differentiating between sets and classes is really an issue of extensionality. --Liempt 06:51, 21 September 2007 (UTC)

Actually, I do think differentiating between sets and classes is an issue of extensionality/intensionality. Oh, maybe not so much for "classes" in the NBG sense, though there's a connection even there, but in the more classical sense that a set is determined by its elements, whereas a class is determined by its membership criterion. So to take the (not entirely accurate) standard example, the set of featherless bipeds is the same as the set of human beings, but the class of featherless bipeds is distinct from the class of human beings.

With the benefit of hindsight we can see that failing to observe this distinction was the real source of the paradoxes. If you're thinking of sets as gathering together pre-existing objects in an arbitrary way, rather than choosing them by a rule, then there is no reason to believe that you can gather together all objects -- the set itself is an object, but is not there to be gathered prior to its formation. So what you naturally get, leaving out urelements, is the von Neumann hierarchy, which is not vulnerable to the antinomies.

The current treatment of the paradoxes in the set article is seriously flawed; the paradoxes do not come from the fact that "well-defined" is not well-defined (though that's a good resolution for a different paradox, the one about the least undefinable ordinal), but rather from the attempt to treat sets as stand-ins for their definitions. This is somewhat of a global problem that I haven't gotten up the gumption to try to fix yet (it's a huge job). See my remarks in talk:naive set theory. --Trovatore 21:22, 21 September 2007 (UTC)

Hey Mike, good points. I was thinking about classes in the way Neumann described them, so differentiating between sets and classes didn't really seem (at least in any strong sense) as an issue of extensionality. While failing to observe this distinction certainly contributed to several paradoxes, I'd say a number of paradoxes were caused by a set's membership criterion being an antimony. I put a great deal of thought into the issue of antimonies and I came upon several interesting results, including a weakening of the incompleteness theorem, which I'll set forth in few months in my doctoral thesis, but wikipedia isnt the place for original research (although if you're interested in Godel's work and foundations, I'd be happy to send you a copy once it's done, I daresay it's quite beautiful), but alas, I digress. Anyhoo, if you'd like to work on the issue of paradoxes within set theory on wikipedia, I'd be happy to help. Give me a shout on my talk page and we'll see what we can do. In the meantime, I'll add a bit on the distinction of sets and classes. One last thing, is the Neumann hierarchy really appropriate to an article of this scope? Note that I am still new to this wikipedia thing and am having trouble with the appropriate level of depth. Cheers, Liempt 04:44, 23 September 2007 (UTC)

I would say the von Neumann hierarchy, considered on its own, is probably not appropriate to the level of this article. However if the antinomies (watch out for the spelling, before you submit your dissertation to your committee!) are to be presented, then I think we may have to mention it. I'm not happy at all with the uncritical presentation of the idea that the problem was an intuitively based set theory and the resolution was a formalistic axiomatization. My view is that the problem was wrong intuitions, and the resolution was getting the right ones. A balanced NPOV presentation is going to be tricky. --Trovatore 07:34, 23 September 2007 (UTC)

Darn it anyway; I do that all the time. Chemistry (I know, I'm ashamed I took it) got antimony into my head and now I mix the two up all the time. My question to you is, how in the world are we going to introduce the concept of a von Neumann hierarchy to an audience for whom a cartesian product is nearly too advanced? —Preceding unsigned comment added by Liempt (talk • contribs) 07:59, 23 September 2007 (UTC)

A History Of Set

It is now time to give a brief review on the history of the article of set in wikipedia. At the very beginning, set looked like a truncated version of Set (disambiguation), with a short introduction of what a mathematical set is. Later we decided to add a simple section about operations on set, indeed it was just a very very short section - union, intersection, subset, and everything else are explained in one or two sentences plus atmost a Venn diagram. Of course there were not really "we", it was just the collective outcome of individuals. We should possible stop there as everything else was already in naive set theory. Somehow, a section of paradoxes appears, and then one by one each set operation has its own section. At the same time the number of different meanings of set grows. Then finally we have the new Set (disambiguation) and the current, still changing set. Hey! Is that any simple way that I can have a look on the history of set? It is a really good research subject! :P wshun 01:06, 26 September 2007 (UTC)

"formal" definitions

Wouldn't it be good to have a few clear-cut definitions of notions such as union and intersection.

For example write $A\cup B=\{x:x\in A\land x\in B\}$. Likewise have $A\cap B=\{x:x\in A\lor x\in B\}$. (replacing maybe the ^ and v by AND and OR, but I don't know how to do that).Randomblue 16:39, 26 September 2007 (UTC)

Actually, it would be $A\cup B=\{x:x\in A\lor x\in B\}$ and $A\cap B=\{x:x\in A\land x\in B\}$.Daniel Walker 04:22, 28 September 2007 (UTC)

Hey guys. I actually added some slightly more advanced propositions earlier and it was decided against. You can read my convo regarding them above. Liempt 05:26, 28 September 2007 (UTC)

Provided Link

Just a newbies take on the content of the link provided, please ignore if inappropriate. The content here - http://www.c2.com/cgi/wiki?SetTheory, is messy, very messy. Surely some higher quality links can be provided here? Just a thought. Jashwood 22:32, 4 October 2007 (UTC)

Images

Quick suggestion from an unregistered viewer. I wanted to bring attention to the images that demonstrate the definitions of union, intersection, complement, and the like. These images do not do a good job of pointing out the definitions. The union and intersection images are extremely similar, with very little showing the distinction between the two concepts, and the images have inconsistent coloring schemes.

My proposal is to redo the images to show more clearly what the resulting value is and to be consistent. Something as simple as circles with a very light gray fill to denote the sets and a light red shaded region to denote the result of the operation would do. It would be better than what is there now.

Thank you. 76.104.25.105 18:33, 7 October 2007 (UTC)