Talk:Set (mathematics)

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Topics from 2002-2006[edit]

Preserved histories[edit]

Set (mathematics)[edit]

   * (cur) (last) 22:23, 17 September 2008 Redirect fixer (Talk | contribs | block) (32 bytes) (Set has been moved; it now redirects to Set (mathematical).) (rollback | undo)
   * (cur) (last) 17:35, 18 April 2002 AxelBoldt (Talk | contribs | block) (Moving back to Set) (undo)
   * (cur) (last) 12:12, 9 April 2002 Zundark (Talk | contribs | block) (moved from Mathematical_set) (undo)
   * (cur) (last) 23:20, 2 April 2002 Toby (Talk | contribs | block) m (undo)
   * (cur) (last) 00:34, 27 March 2002 (Talk | block) (Russell's paradox) (undo)
   * (cur) (last) 00:33, 27 March 2002 (Talk | block) (Russell's Paradox) (undo)
   * (cur) (last) 10:21, 26 March 2002 -- April (Talk | contribs | block) m (move math useage here)


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


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

{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)

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.

list v. set[edit]

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)

Set vs. Naive set theory[edit]

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.


Paul August 20:23, Aug 16, 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, Aug 27, 2004 (UTC)

∪ symbol displays as box?[edit]

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, Aug 31, 2004 (UTC)

symbols displaying as boxes[edit]

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

Symbols and the set theory[edit]

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 Sep 2004 (UTC)

School curricula[edit]

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)


User: 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:
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?
A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?
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?
A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.
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'

if vs iff in mathematical definitions.[edit]

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)

Symbols displayed as boxes[edit]

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


I didn't think this was controversial, but when 95% of our readers use standard versions of IE with no extra fonts, I think it is us who should be conforming to their needs. IE does support empty set and intersection characters, but not union or subset. I've replaced any union characters with a capital U (which looks pretty good in a sans serif font) and I replaced some subset symbols with "is a subset of", and others with images using math tags. I'd advise similar practices be followed in other articles. Note that if you set your preferences to "HTML if possible", the formulas in math tags should be rendered correctly as HTML for people with browsers supporting all these characters. Deco 20:56, 11 July 2005 (UTC)

Box Fixing Redux[edit]

Someone PLEASE tell me we can do better than peppering the article with images to fix the "symbol/box/font" problem! Unicode should be ubiquitous by now. Can not everyone read these symbols? ⋂ ⋃ ∈ ∉ (talkcontribs) 15 January 2008

Hello. I am using IE 7.0, and all of those symbols appear as boxes.
--Bob K (talk) 14:32, 19 January 2009 (UTC)
Eureka! I just changed Times New Roman to Lucida Sans Unicode in a pull down menu at Tools -> Internet Options -> Fonts -> Webpage Font:,  and now the symbols work.
--Bob K (talk) 14:49, 19 January 2009 (UTC)

Some reverted edits[edit]

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, Jan 7, 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, Jan 9, 2005 (UTC)


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, Apr 25, 2005 (UTC)

Removed inappropriate (in my view) text[edit]

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, Apr 29, 2005 (UTC)

Living dragons[edit]

How do we know that set A is equal to the null set (where A is the set of living dragons)?

Lead section[edit]

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, July 11, 2005 (UTC)

My reversion of edit to "definition" section[edit]

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. 10:46, 22 August 2006 (UTC)


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"[edit]

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


SECTION: Cardinality of a set[edit]

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? 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 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)


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)

Improper Subsets[edit]

Can we get a clarification on Improper Subsets? —The preceding unsigned comment was added by MLeg11 (talkcontribs) 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)

Topics from 2007-2008[edit]

Contradiction / clarification?[edit]

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[edit]

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:
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
"is a subset of"; "is a subset, or equal to"; "is a superset of"; and "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[edit]

1={1} ?

{{1}}={1} ? 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)


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

How many elements?[edit]

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 2R, 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)

More formal definitions of Union, Intersection, Complement, Subset[edit]

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

  • Subset -
  • Union -
  • Intersection -
  • Complement -

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 . 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?[edit]

(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)

No mention of extensionality?[edit]

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?— 02:11, 21 September 2007 (UTC)
Son of a gun! I'd timed out when I signed above. For completeness, 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 (talkcontribs) 07:59, 23 September 2007 (UTC)

Cartesian Product[edit]

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)

A History Of Set[edit]

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[edit]

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

For example write . Likewise have . (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 and .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[edit]

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


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. 18:33, 7 October 2007 (UTC)

Venn diagrams[edit]

I changed some graphics in the article from files like this Venn A intersect B.svg into files like this Venn0001.svg.

Using several (maybee more beautiful) colours, the old diagrams are less clear than those simply using white and red. The reality knows only two possibilities: An area belongs to the set or not. There is nothing in between, and the diagrams should reflect that.

All the more, most of the old diagrams lack the margin standing for the universe. For a blackboard scetch this might be acceptable, but not for an encyclopedia. A set diagram without a universe is simply false - not only for sets containing the intersection of both complements.
--Tilman Piesk (talk) 20:48, 2 February 2008 (UTC)

Apart from the original slides being prettier, I find them more clear since the background and foreground have different colors. Oleg Alexandrov (talk) 01:02, 3 February 2008 (UTC)
This discussion is based on a unfortunately common confusion between the work of Leonhard Euler (viz., the Euler diagram) and the work of John Venn (viz., the Venn diagram).
Euler diagrams, in every application, have no "margin standing for the universe".
By contrast, the defining and specific feature of a Venn diagram is the presence of "margin standing for the universe" (it must be understood that, historically, Venn diagrams are Euler-diagrams-as-improved-by-Venn).
Thus, any claim that a set of circles, standing alone, with no surrounding margin, is a "Venn diagram" is not only plainly wrong but is, also, a sign of complete ignorance of the history, significance, graphic conventions, and the application of such diagrams -- all of which were, in their origin, attempting to improve on the conventions for graphical representations of logical reasoning statements that had been first proposed by Gottfried Leibniz. Thus, it is obviously imperative that:
(1) Anything labelled Venn diagram must have a "margin standing for the universe"; and
(2) Anything that does not have a "margin standing for the universe" must be labelled a Euler diagram. (talk) 02:16, 3 February 2008 (UTC)
Is this A\B? Is A on the right? Venn0010.svg
Since "relative complement"/"set-theoretic difference"/"subtraction" is not a symmetric, it is important to label A and B. To me, the graphic intuitively looks like B\A, not A\B.
mjk (talk) 10:01, 25 January 2010 (UTC)
I've changed to the A\B diagram and changed the comment below so it goes with the text beside it. Dmcq (talk) 12:58, 25 January 2010 (UTC)

Set as an abstract object[edit]

I disagree (rather strongly) with this edit. A set is indeed abstract, but most people don't know what "abstract" is mathematically, and if anything, the concept of set is abstract enough (the irony) that we better don't mention that very thing in the first sentence.

There's been a long argument between me and Pontiff Greg Bard on my talk page about this (User talk:Oleg Alexandrov#a set is an abstract object) with a third party, JackSchmidt, putting in a third opinion which I will quote below. Other comments?

(The comment below is originally from User talk:Oleg Alexandrov.)
(Hopefully I am not butting in.) The edit appears fairly minor, and probably the two positions can be merged easily. I can agree that "a set is an abstract object" is one of the first things we learn about sets, since they are quite often once of the first "abstract" things you study in math (using mathematical language here). I think the part of the edit that makes the article less simple is linking to rather refined, though fundamental, philosophical articles in the lead of an article on a mathematical subject. Since set theory, especially naive set theory, has caused quite an examination of the philosophical foundations of mathematics, such links are appropriate *somewhere* in the article. I think the only concern is that it not be in what is more or less the opening sentence.
Could both agree to "In mathematics, a set is an abstract object that can be thought of as any collection of distinct objects considered as a whole." as the opening sentence (with only the single wikilink), and then an addition, perhaps to the Definition section (so quite fundamental to the article, and appearing quite early), links to the philosophical concepts being used somewhat informally there? I think the tone in that section is quite suitable for wiki-links to the philosophical articles that provide a sounder understanding of the terms. Alternatively one could follow a somewhat common style of having a two or three paragraph lead that has sentences for what it is, what it is used for, its history, its foundation, and its generalizations. I think a very short lead is much nicer, and that editing in the Definition section would probably reach consensus more quickly, but I don't see any insurmountable trouble with expanding the lead. JackSchmidt (talk) 06:08, 5 February 2008 (UTC)
I still disagree the first sentence of the article is the place to settle such fine points. Oleg Alexandrov (talk) 06:24, 5 February 2008 (UTC)
I hope the proposed solution does not seek to settle such fine points in the opening sentence, merely it provides text both parties can agree on. I was hoping that the word "abstract" itself is not a problem, merely the attempt to say specifically what we mean by this word in the opening sentence.
Apparently the word "abstract" is not considered a general-service word, wiktionary:simple:abstract, but I think we were taught this word in k-6th grade, but certainly in 7-12th. Again hopefully we can agree the word itself is not a problem, merely being too formal about the word in the first sentence is worrisome to Oleg Alexandrov and others, and not fundamentally important to Greg Bard and others. The addition to the definition section should provide philosophical context for "object", and if any fine points need to be settled, there should be a section on such things. JackSchmidt (talk) 06:55, 5 February 2008 (UTC)
I am not so concerned with object (philosophy) as a link. It is there so as to distinguish from the mention of "abstract object" (and it was working well in that capacity). It was linked because it was an appropriate wikilink to a term which was already in the article, which would benefit the article as a wikilink. My main concern is calling something what it is. I don't think we need to point out in every biography that "Abraham Lincoln, was a human being, who..." Because, although that fact is equally fundamental to the concept of Lincoln, it is also sufficiently obvious to everyone. For such a concept as set, I would think that we would want all the tools at our disposal to say what it is on that fundamental non-obvious level.
Oleg, please observe: "most people don't know what "abstract" is mathematically," Where does the idea ever come from that every single point about the thing has to be "mathematical"? I already told you that mathematicians do not have the monopoly on the study of this thing. The first step to working interdisciplinarily is admitting that we have an interdisciplinary subject. Please ease up on the grip over these type of articles. Your statement also supports the thesis that it should be elucidated upon, not omitted.
Abstract objects are not made more abstract by identifying them as "abstract objects." They are made less abstract. This would seem to be a self evident principle (about language, not set). I am sorry Oleg, but it is on this ground that I will be reinserting that phrase. Please reconsider your opening remark above.
Where does the idea come from that it might be okay to have this as the first sentence, but it would be better to wikilink abstract object somewhere else in the article? The guideline on wp says that the first instance of a term should be wikilinked, but not the others. So exactly why would this be an exception? It is an exception which is consistent with all of the rest of the de-emphasizing, mitigating, and outright denial of the logical (metalogical/metamathematical/philosophical,etc) foundations of these concepts which I am endeavoring to insert. It is a POV issue, and a I wish the mathematicians were more conciliatory, rather than hostile to these foundations (which invariably, they don't find important at all.) Be well, Pontiff Greg Bard (talk) 17:23, 5 February 2008 (UTC)
Every mathematical object is abstract, so saying that any particular one is abstract is unnecessary. Worse it is misleading since it gives the false impression that some mathematical objects are concrete. I strongly support the current formulation, over the proposed one. Paul August 18:33, 5 February 2008 (UTC)
Paul, the last sentence says that set theory can be viewed as foundational to the rest. That means we do not need to identify every single abstract object as such --just the most important ones. Please reconsider. Please consider what I have said above about the hostility to these concepts. Why does everybody feel so strongly? I find that troubling, and closed minded. Be well. Pontiff Greg Bard (talk) 19:02, 5 February 2008 (UTC)
Saying that a set consists of objects, with object linking to object (philosophy), does seem rather confusing since that article says that object has different meanings, including some that are simply wrong in the context "a set consists of objects". Similarly, I don't see how a link to abstract object helps the reader to understand what a set is. Saying that a set is an abstract object, while its elements are objects, is confusing because it implies that a set is more abstract than its elements. Greg, I think that what you're calling hostility against logical/philosophical ideas is in fact hostility against your logical/philosophical ideas. -- Jitse Niesen (talk) 20:21, 5 February 2008 (UTC)
I tried to find a compromise between the two positions. I think my text is slightly better than the previous one and Gregbard's, but like Jitse I find the links misleading. We could, perhaps, replace the two occurrences of "object" in the sentence by "thing". I think this would give the wikilinks a context in which only philosophically minded people would click them; which seems to be exactly what we need here. --Hans Adler (talk) 20:32, 5 February 2008 (UTC)
I think that the original version is better — and so I've restored it per Hans' edit summary. Paul August 20:36, 5 February 2008 (UTC)
Can you explain what you didn't like, so we can move towards a consensus? Somehow I feel that my version was slightly clearer, but I can't really say why. (I am not talking about the links here.) --Hans Adler (talk) 20:42, 5 February 2008 (UTC)
In my opinion it is simpler, more elegant, easier to understand, and as I said above, does not contain the unnecessary and misleading term "abstract". Paul August 20:47, 5 February 2008 (UTC)
I think I see now what I like about my version: It starts with the problem. We have a bunch of things, and we want to do mathematics with them. So we just pretend that all the things together are one thing. The old version sounds as if mathematicians had discovered sets lying around somewhere, not knowing what to do with them, and then one day a bright guy discovered that you can think of them as just a collection of objects. This seems to make them more mysterious. --Hans Adler (talk) 20:58, 5 February 2008 (UTC)
Moreover, "In mathematics, a set can be thought of…" sounds as if it should be continued with something like "whereas in law, a set is…". My point here is that we seem to have developped Wikipedia jargon for this kind of situation, in which we stress the fact that the word has several meanings a bit more than we should. --Hans Adler (talk) 21:59, 5 February 2008 (UTC)

Perhaps it helps to have all the versions together here. Feel free to add new versions.

  1. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. [Traditional version]
  2. A set is an abstract object. It can be thought of as any collection of distinct objects considered as a whole. [Gregbard's version]
  3. In mathematics, any collection of distinct objects can be treated as a new abstract object, called a set. [Hans' version]

--Hans Adler (talk) 20:49, 5 February 2008 (UTC)

The question whether sets are abstract or not is somewhat subtle; see Penelope Maddy, and Realism in Mathematics (p. 2) on Google books. As it stands, the article is essentially a fork of naive set theory. I would rather see this article cover sets more broadly, including philosophical issues such as concreteness. I am not in a good position myself yet to write that material here, though. — Carl (CBM · talk) 00:15, 6 February 2008 (UTC)
Going beyond naive set theory is a worthy goal. However, starting the article with a rather obfuscated first sentence (sorry Greg and Hans) is probably not the way to go. Adding another short paragraph in the introduction about deeper issues (without getting too technical) could be a good idea. Oleg Alexandrov (talk) 04:02, 6 February 2008 (UTC)
I agree that the first sentence is not the place for fine points. What about simply "In mathematics, a set is a collection of distinct objects." — Carl (CBM · talk) 04:46, 6 February 2008 (UTC)
I have to concur with the position that the introductory sentence should not be intimidating and should not be the result of a complex compromise between mathematicians and philosophers. If need be, a sentence can be added to point out that formal definitions of the notion can be tricky but the development of these ideas should be explained later in the article. Linking to Object (philosophy or abstract object is a bad idea. Most readers come in thinking they have a good intuitive idea of what a set is and are told from the get go "you're not even close, you moron". Pichpich (talk) 19:30, 6 February 2008 (UTC)
When I tried reading David Hume's A Treatise of Human Nature for the first time at age 16, right from the "get go" I got the message "you're not even close, you moron." This turned out to be one of the most intellectually edifying experiences in my life, forcing me to new educational heights. Some subjects are just plain harder than others; and for many people, higher mathematics is one such subject.
That said, I like Hans Adler's suggestion best from the list above, though we might also consider:
  • In mathematics, a set is any abstract object consisting of a collection of distinct (abstract or concrete) objects.
However, Carl raises a good point about the status of realism in mathematics. While I may think that such realists are misguided, it is surely not a settled issue. And if they are correct, then no mathematical objects are abstract (unless, of course, a mind independent realm of mathematical forms would still count as abstract in a relevant sense—a very real possibility). Given this, we might consider:
  • In mathematics, any collection of distinct objects can be treated as a new object, called a set. [rewrite of Hans' version]
  • In mathematics, a set is any object consisting of a collection of distinct (abstract or concrete) objects. [rewrite of my version]
Regardless, I find the "traditional version" unnecessarily awkward, and I think it would be appropriate to replace it. Postmodern Beatnik (talk) 16:23, 13 February 2008 (UTC)
Try to think from the point of view of the reader who never heard of sets before. Tell him in plain language what a set is. Trying to go to the fine points before even giving the reader a chance to get an idea about the topic is not good I think. Oleg Alexandrov (talk) —Preceding comment was added at 03:35, 14 February 2008 (UTC)
Well, I'm not a mathematician and I'm not confused by calling a set an abstract object. Indeed, I doubt the concept of being an abstract object is too difficult for a reasonably intelligent person. And the lead section is supposed to sum up the article. If the bottom line on sets is that they are abstract objects consisting of a collection of other objects (either abstract or concrete) then that's what the lead section should say. Postmodern Beatnik (talk) 20:21, 19 February 2008 (UTC)
You're trying to put too many ideas in one sentence. Again, that's too much to swallow to somebody who never talked about sets before. You're very welcome to add a second paragraph to the introduction elaborating on the abstractness issue, it just should not be in the first sentence. Oleg Alexandrov (talk) 05:23, 20 February 2008 (UTC)

I have given ten diverse sources that identify a set as an abstract object. There are sources here from mathematical logic, and various other disciplines which would seem to elevate the appropriate level of emphasis.

  • Identifying a particular abstract object and calling it an "abstract object" helps us understand something fundamental about the object, and makes it possible for us to talk about it intelligently.
  • There is nothing "misleading" about calling a set an "abstract object." The opposite is true: we are leading people to think appropriately about this concept by identifying it as such.
  • The "set" is probably the most famous "abstract object" there is. In almost any account of what is an abstract object, the "set" is usually given as the first example, or portrayed as the paragon of an example of such.
  • At least one of these references opens a chapter "A set is an abstract object."

I hope we can see our way to working with it. Be well. Pontiff Greg Bard (talk) 10:23, 4 March 2008 (UTC)

As I pointed out above, those who advocate realism in mathematics (such as Penelope Maddy did, at least at one point) say that sets are not abstract objects, but concrete ones. The abstract/concrete nature of sets would be well served by a paragraph in the article, but not by a single sentence in the lede that glosses over the issue or treats it as settled. Also, I'll point out that there are very few other mathematics articles that begin by pointing out that the obejcts at hand are abstract. We don't see:
and so on.
I started that section on the nature of sets; I'm sure it can be filled out some more. I changed the lede to refer down to that section, and to point out that the issue of abstractness is not completely clear. — Carl (CBM · talk) 14:25, 4 March 2008 (UTC)
The lede looks all right. I'm just glad this is in there pretty close to the start. In the case of these others you mention, I would point out that not every abstract object needs to be explicated in this way, however some of the most important ones should be. In the case of ring I notice that there are wikilinks as follows:
ring --> Algebraic structure --> set --> abstract object
This type of arrangement is fine with me. This way people have a chance of nailing this down if they explore deeply enough. Pontiff Greg Bard (talk) 00:12, 5 March 2008 (UTC)
The thing is looking pretty good in my view. My only notes are the following: "In philosophy" implies that there is some world of philosophy that is separate from the rest of the world. This is a general complaint I have about WP. If it is supposed to be true in one field or another it is supposed to be true for all rational beings (especially in logic!) I think we should find some other way to insert a wikilink to mathematics, or logic or whatever; otherwise it just seems like a gratuitous way to insert "philosophy" into the article which I don't see as necessary. The other issue I am observing, but in which I do not have any special knowledge is the Maddy position. It doesn't say in her article that she believes sets are concrete, it says "can be." I was not able to make the stronger statement as it is worded now because of that. If you are sure, then that's that for me Carl. Are you absolutely sure? Her position doesn't make complete sense to me as presented (but that's ok). Be well, Pontiff Greg Bard (talk) 23:57, 12 March 2008 (UTC)
A key role of the lede is to establish context. The term "abstract" is not used here in the naive English meaning, but the specialized meaning from philosophy. Compare other mathematics articles, for example center (group) which starts out "In abstract algebra..." even though abstract algebra is the only setting in which the center of a group is studied.
I seem to remember some of Maddy's book is on google books. I am no expert in her philosophy, but I can get the book from the library and look at it. — Carl (CBM · talk) 01:40, 13 March 2008 (UTC)
Well I don't see this "set" edit as the beginning of a revolution in rearranging every lede of all the articles that begin that way. However, I think we should use that format in some places, and not others. "In food science, a sandwich is ..." would be ridiculous. I think other people enjoy sets other than mathematicians. So that's my perspective on why I think it's an issue. Pontiff Greg Bard (talk) 03:56, 13 March 2008 (UTC)
Constructions like "In rocket science, lettuce is any green plant other than eruca sativa" are a bit strange, but then large bodies of text like encyclopedias are allowed to have some idiosyncracies. In this case the qualifier "in philosophy" is appropriate because when people think about whether sets are abstract or concrete they are really doing philosophy, and without the qualifier readers might think that mathematicians (as such) are concerned about this. They are not. I think most of us are not interested in such points at all. This is not something we teach, or read or write about, and at least in my experience it is also not something we talk about at parties. I think raising the issue in the lede of set (mathematics) at all gives it undue weight, but I don't object so long as it is properly qualified so that people know they needn't look up the meaning of "abstract" just to understand one of the most fundamental notions in mathematics. --Hans Adler (talk) 08:18, 13 March 2008 (UTC)


User:DionysosProteus has rather boldly moved set to set (mathematical) and redirected set to set (disambiguation). Now I admit it's not completely implausible that set should be a disambiguation page, but I think it was a bit excessive to suddenly move a long-standing article with no discussion.

Anyway here are my specific concerns:

  1. Yes, set has a huge number of meanings, but most of them are rather marginal as topics for an encyclopedia article. I'd go so far as to say that the mathematical meaning is the one, by an enormous margin, most worthy of an article. This is a bit reminiscent of the QED debate — quantum electrodynamics is not the meaning most known to most people, but it is by far the one most important to treat in an encyclopedia article.
  2. Even if the move were correct, the name set (mathematical) is wrong; by WP convention it would be set (mathematics).
  3. Similarly, if set is to point to the disambiguation page, then the disambiguation page should simply be called set, not set (disambiguation). --Trovatore (talk) 22:35, 17 September 2008 (UTC)
I was bold? Wikipedia told to be so. Points 2 & 3 are fine but #1 isn't. Sets on films and in theatres, the sets of a tennis match, and a DJ's set are not marginal meanings. I would think it's doubtful that something like a "head-count" of instances in which most people use the word in everyday life would bring its mathematical usage out on top. Not sure by what critera the mathematical meaning is more significant for an encyclopedia than these cultural definitions. Mathematics is more "important" than Drama or the Cinema? Not sure what evidence would settle the question either way. I should think that the first line of the disambig page: "Set has 464 separate definitions in the Oxford English Dictionary, the most of any English word; its full definition comprises 10,000 words making it the longest definition in the OED" should argue for a disambig at Set. DionysosProteus (talk) 22:45, 17 September 2008 (UTC)
I think Trovatore's point was that it's not everyday usage that matters, it's encyclopaedic value. For example, no-one would write an article on tennis sets (we have one on tennis scoring which discusses the concept) even though they are probably talked about more often than mathematical sets. --Tango (talk) 22:50, 17 September 2008 (UTC)
Everyone thinks that WP:BOLD is a license to try anything. Fewer have actually read the guideline. Please read it. Geometry guy 23:05, 17 September 2008 (UTC)
It pretty much is a license to try anything, as long as you don't mind people reverting you. --Tango (talk) 23:17, 17 September 2008 (UTC)
So you haven't read it either, then. Geometry guy 23:22, 17 September 2008 (UTC)
For a radical edit, the bold-revert-discuss protocol would be fine here. The move ought to be reverted pending discussion (even though, as I say, it's not implausible that the discussion would come out in favor). But now that can't be done without admin-deleting set. That's why WP:BOLD specifically mentions page moves as a place to be careful. --Trovatore (talk) 23:24, 17 September 2008 (UTC)

Support move but fix up with proper names: Set (mathematics) for maths version and Set for disambig page. I don't think there is an overwheling case for the mathematical meaning being overwhelmingly the most significant. Last month views were Set 28748, Set_(mythology) 18911. It will be a pain to fix all the redirects, but I can't see a long term problem. --Salix alba (talk) 23:38, 17 September 2008 (UTC)

I was skeptical of this move, but I'm willing to entertain the possibility that the mythological meaning is common enough to support make the primary page a dab page. Dcoetzee 23:43, 17 September 2008 (UTC)
Given the large number of meanings of set, I'm willing to buy the arguments thus far...but let me ask Salix alba to clarify his. Apple Inc. gets a comparable number of page views (83,500) compared to Apple (116,000); note this is a smaller disparity than pointed out for the examples of set. Should Apple thus become a disambiguation page? --C S (talk) 01:35, 18 September 2008 (UTC)
That is a good point. I think the argument in that case goes the other way. Between apple and Apple computer, it's clear that the fruit is the "principal" meaning: the computer company is named after the fruit. No such argument applies in the case of mathematical sets, tennis sets, the Egyptian god of the dead, etc.
Furthermore, the principle of least surprise argues for apple to refer to the fruit. A reader who arrives at apple hoping to find Apple Computer but who finds red fruit instead will instantly understand exactly what happened. But someone getting a mathematics article when they wanted the Egyptian god of the dead (or vice versa) is more likely to be puzzled and surprised.
I don't know what Salix Alba's idea was, but that's how I would argue it. I hope this has been of some value. -- Dominus (talk) 03:55, 18 September 2008 (UTC)
It has certainly convinced me that making Set a disambiguation page was a good idea. (Even though I think it was done a bit too boldly.) --Hans Adler (talk) 13:20, 18 September 2008 (UTC)
The "Principle of least surprise" is something I can understand in establishing a "primary" meaning or lack of one. It's certainly a good argument to use here as in the case of apple and many others. That's why I'm willing to "buy" the move, even if the "primary" is just a wiktionary entry, that may be less surprising than being sent to a mathematics page. A disambiguation page certainly makes sense here. I think Salix Alba's page view argument, however, is a red herring. Just like the large number of people viewing Apple Inc may first find they are directed to Apple, it may be that the large number viewing the mythological set are fully aware why they were first sent to a mathematical article. Surprise is in the eye of the beholder. Perhaps it's natural to think the mathematical set is the least surprising of all the Wikipedia entries to be sent to, but I can't help but think most people would view a "collection" or disambiguation as the most natural result. --C S (talk) 10:25, 19 September 2008 (UTC)

I've now moved Set (disambiguation) to Set.--Salix alba (talk) 17:47, 19 September 2008 (UTC)

This has created about a thousand incorrect links. Please fix them. (And next time, please fix them before doing the move.) --Zundark (talk) 18:11, 19 September 2008 (UTC)
I'm going through them now, folks can help if they want (see AWB). Quite a few did not really want to link to Set (mathematics), many just want to link to the more general English word for collection rather than a more technical meaning and others want Set (computer science), a "set of cards". Philisophical and linguistic uses are the trickiest. --Salix alba (talk) 23:08, 19 September 2008 (UTC)

Topics from 2009-2010[edit]

It never ends[edit]

Quote from NBG theory:

Category theory

The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. In some developments of category theory, for instance, a "large category" is defined as one whose objects make up a proper class, with the same being true of its morphisms. A "small category", on the other hand, is one whose objects and morphisms are members of some set. We can thus easily speak of the "category of all sets" or "category of all small categories" without risking paradox. Those categories are large, of course. There is no "category of all categories" since it would have to contain the category of small categories, although yet another ontological extension can enable one to talk formally about such a "category" (see for example the "quasicategory of all categories" of Adámek et al (1990), whose objects and morphisms form a "proper conglomerate"). On whether an ontology including classes as well as sets is adequate for category theory, see Muller (2001).


The 3 main paradoxes of set theory (Cantor Burali-Forti Russel's) have caused people to develop axioms to avoid such abominations. However to me, and I believe many other people, it still seems appropriate to use unrestricted comprehension on collections. So for this matter it is sensible to talk about the "hyper-set" of all sets (which is simply a dupe for class), and "hyperclass". If we take the empty set in von Neumann's universe construction to be the von Neumann universe, and naively assume "power collections" exist (I'm not sure if this can be put into an axiom), it's highly intuitive to think of the "2nd von Neumann universe" as the hyperclass of all classes. Now we don't have to end here. Indeed for any successor ordinal α we can define a hyper-α-class from the previous ordinal, and for limit ordinals define the hyper-α-class to be the union of all hyper-β-class where β<α. What we have now is a von Neumann universe of hyper von Neumann universes. Again, why stop here? We can keep going and going and whenever you suggest something that contains "everything" we can go further... It never ends. Any comments?--Standard Oil (talk) 03:48, 20 February 2009 (UTC)

The talk page is for discussion about improvements to the article (see WP:TALK). This material is not suitable for inclusion because it is not the subject of a reliable, published source on set theory (see Wikipedia:No original research). Dcoetzee 03:54, 20 February 2009 (UTC)

Which is it?[edit]

These sentences:

For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

Seem to be in contradiction. The first asserts that |N| is less than |R|, but the second seems to explain why that wouldn't be the case because the cardinality of a segment of a line is the same as the cardinality of the whole line. Am I reading this wrong? —Memotype::T 00:00, 12 March 2009 (UTC)

Every segment of a line that is nontrivial in the sense that it contains more than one point has the same cardinality as R. Did you confuse "segment" with "sequence"? --Hans Adler (talk) 00:45, 12 March 2009 (UTC)
No, but what you just said contradicts the assertion that the set of real numbers has greater cardinality as the set of natural numbers — both are equally infinite, no?. —Memotype::T 03:56, 12 March 2009 (UTC)
No. Where do you see a contradiction in what Hans said? The set of natural numbers is not a segment of the real line, so the fact that every nontrivial segment (interval) taken from the line has the same cardinality as the line itself, does not contradict the fact that that cardinality is strictly greater than the cardinality of the naturals. --Trovatore (talk) 07:42, 12 March 2009 (UTC)
The cardinality of an infinite subset of the line has nothing to do with whether it stretches infinitely in one or both directions. The only thing that matters (for the kind of tame subsets we usually have in mind) is whether it contains a full segment. In fact, for any such segment it's easy to transform it into the full line, in such a way that for every point in the full line there is exactly one point in the segment. --Hans Adler (talk) 08:43, 12 March 2009 (UTC)
I kind of see, but I'm still a bit confused. When we say "cardinality" we mean the number of members in a set, right? The number of members in the set of natural numbers is infinite, and so it the number of members in the real numbers set, no? Thus, how does it make sense to say the cardinality of natural numbers is less than the cardinality of real numbers? —Memotype::T 22:47, 12 March 2009 (UTC)
Both are infinite, but the set of reals is a bigger sort of infinite. A lot of people have a preconceived notion that there's no such thing as "a bigger sort of infinite", but there is, there really really is. You have to unlearn that prejudice and move on. --Trovatore (talk) 00:59, 13 March 2009 (UTC)
Actually, I don't agree with that because it's a platonist statement. The "bigger sort of infinite" is only a fiction that is convenient (perhaps even necessary) for mathematicians. It doesn't really reflect any phenomena that we can observe in the real world; after all, so far as we know the universe is really finite, i.e. it has finite diameter and contains only a finite number of particles etc. In the real world, even a line segment contains only finitely many distinguishable points. But if we want to talk about all natural numbers, we need infinity, and if we want to talk about all sets of natural numbers we need the "bigger sort of infinity". Without being able to talk about these things we mathematicians would be deprived of most of the beautiful objects that we are studying. --Hans Adler (talk) 02:27, 13 March 2009 (UTC)
I am, of course, speaking as a mathematical realist (sometimes imprecisely called Platonist). --Trovatore (talk) 05:15, 14 March 2009 (UTC)
Perhaps it would be better to say "it's a bigger infinite cardinal", which is an extremely well defined concept, unlike "infinity". It's far easier just to say "infinity" and assume people can work out what you mean from context, though. --Tango (talk) 21:27, 14 March 2009 (UTC)

Determinate member?[edit]

Is there such a thing? I expect there is, and I've got the name wrong. What I mean is a set with various members, but with one dominant (is that the word) member, a member which if taken away means the set no longer exists, even if all the rest are still there. Is such a collection then in fact a set? I'm thinking of the Solar System which is defined as, "The Solar System consists of the Sun and all those objects bound to it by gravity". Which, I think, is an intensional definition. Of course if you take the Sun away, that is the same as taking everything else away, and the set still exists - it's just the Sun with an empty set of objects orbiting it. Anyway, you can see I don't usually work with any mathematics, but I hope I have enough intuition to understand simpler answers on this!! HarryAlffa (talk) 15:46, 8 May 2009 (UTC)

The term doesn't exist and the concept makes no sense. Removing any member of a set always gives a set as a result. Examples: {a,b, c} \ {a} = {b, c}, {a} \ {a} = the empty set. Removing the Sun from the set consisting of the Sun and all objects bound to the Sun by gravity, gives the set of all objects bound to the Sun by gravity. Paul August 20:00, 8 May 2009 (UTC)
I thought I'd pretty much expressed the same thing on removing members from the set - or at least tried to! The Intensional definition is, "The Solar System consists of the Sun and all those objects bound to it by gravity". What can be said of the Sun in this set, besides being a member? I think this is the question behind my ignorant musings. HarryAlffa (talk) 20:25, 8 May 2009 (UTC)
In talking of the Solar System I'm wondering if saying, "The Sun determines the Solar System", means what I think it means - and can I link the word "determines" to a nice article in the Set Theory category? HarryAlffa (talk) 20:25, 8 May 2009 (UTC)
You are talking about aspects that mathematicians usually ignore; from a mathematical point of view an intensionally defined set is exactly same set as the extensionally defined set with the same elements. (See axiom of extensionality.) It's not at all clear what "taking the Sun away" is supposed to mean on the level of abstraction on which mathematicians work. Presumably you mean you are changing the world somehow, but before you have clarified everything a mathematician can't really say much about your problem. Thinking about the possible clarifications, or pretending that they are unnecessary and trying to get away with it, all of this is in the domain of philosophy, while this is primarily a mathematical article and if you ask a question here you are most likely to get responses from mathematicians. --Hans Adler (talk) 20:45, 8 May 2009 (UTC)
Well, I would quibble with some details. The distinction between extensional and intensional collections is actually quite important to mathematicians — it's precisely the confusion between them that leads to the classical antinomies, such as Russell's paradox. Roughly speaking, sets are extensional (arbitrary lists of things where you don't care about their order or multiplicity), whereas classes are intensional (defined by a rule). Sometimes the intensional notion comes up implicitly, as when you have a code for a Borel set, and for certain purposes you want to consider the sets coded by that code in different models as being "the same", even though they don't have the same elements.
However I agree with Hans that sets as mathematicians think about them don't typically involve physical objects such as the Sun. --Trovatore (talk) 21:13, 8 May 2009 (UTC)
The difference between extensional and intensional definitions becomes important when you have a "functorial" situation in the widest sense, e.g. when you extend or restrict your set-theoretical universe. HarryAlffa's question seems to be related to this, but it's way too vague to be sure. And in any case I am not away of any such notion of "determining", under any name, in set theory (for which I am not an expert, though). --Hans Adler (talk) 21:45, 8 May 2009 (UTC)
There at least I agree with you: I can't think of any way to interpret the phrase the Sun determines the Solar System with a technical set-theoretic notion of determine. --Trovatore (talk) 22:00, 8 May 2009 (UTC)

Ok! Thanks guys! I got interested if there was some concept of the sort I've tried to describe, but this is a side-track really of my real-world first desire of providing an explanation of my use of the word determines in, "The Sun determines the Solar System". What I'm trying to convey here is that it's the properties of the Sun from which all the other properties of the Solar System flow; if the Solar mass was different, then the orbits of all the planets would be different etc. etc. I think that's what my use of determines means here, and I think it's a nice way of putting it. Does it make sense to describe, as an intensional definition, "The Solar System consists of the Sun and all those objects bound to it by gravity"? In short, does anything in Set theory have anything informative to convey to an ordinary reader about these two phrases? Or indeed that they are correct phrases to use? I hope I'm not boring you with this! HarryAlffa (talk) 15:25, 9 May 2009 (UTC)

I think what you are looking for is a map (mathematics) (or "function"). Most non-mathematicians are only used to functions from numbers to numbers, but the concept is much more general. Consider the map f that associates to every object x the set f(x) of all objects that are bound to x by gravity. Then the Solar System is f(Sun). This map makes some of your assumptions explicit. --Hans Adler (talk) 15:39, 9 May 2009 (UTC)
Ah! That sounds like an exact answer for the definition! I will check it out forthwith! HarryAlffa (talk) 18:10, 9 May 2009 (UTC)
You've pretty much described the function above, and I understand that, I'm not sure about linking to function though for the definition, I think the intensional definition would be more accessible to a general reader. HarryAlffa (talk) 18:21, 9 May 2009 (UTC)
Honestly, sorry to argue with Hans, I don't see that it addresses the issue at all. Functions also can be viewed from an intensional or extensional perspective, and they are quite different things: An extensional function, the kind we mostly deal with in set theory, is an arbitrary association between objects, whereas an intensional one is a rule for picking out an object given an object. --Trovatore (talk) 18:24, 9 May 2009 (UTC)

Retrofit talk-page year headers/subpages[edit]

17-Nov-09: I have added subheaders above as "Topics from 2002-2006" (etc.) to emphasize the dates of topics in the talk-page. Older topics might still apply, but using the year headers helps to focus on more current issues as well. The topic-year boundaries were located by searching from bottom for the prior year#. Afterward, I dated/named unsigned comments and moved 1 entry (titled "Images") into date order for 2007 & 2008.
Then I added "Talk-page subpages" at the TOC. -Wikid77 (talk) 19:07, 17 November 2009 (UTC)

Blackboard Bold vs Bold[edit]

I don't know if this is a degeneration led from the web, but I've never seen the noted sets (such as ) represented by anything other than blackboard bold before. Why is mere embolding (such as N) being treated as the preferred font? NathanZook (talk) 02:25, 21 April 2010 (UTC)

Bold alone was the traditional way, actually; blackboard bold was not available in professional typesetting until TeX. In this case, there are two concerns:
  1. Almost everyone can see bold letters in their browser, but people may not have fonts with blackboard bold characters. So we would have to use images to display blackboard bold, which looks worse than just bold. See WP:MOSMATH#Blackboard_bold for more on this
  2. This article is meant for a very general reader, who will probably not be familiar with blackboard bold anyway.
I am sure that we have some Wikipedia articles that use blackboard bold; it's very hard to ensure 100% consistency. But regular bold seems like the best choice here. — Carl (CBM · talk) 02:53, 21 April 2010 (UTC)

Special characters template[edit]

We don't usually add the {{SpecialChars}} template to mathematics articles. This template contradicts the WP:LAYOUT guideline, and to the best of my knowledge no guideline whatsoever even suggests using the special characters template. So I have removed it (for the third time) from this article. If it really truly belongs here, then a special case needs to be made for adding it to this article (as opposed to the many other mathematics articles on Wikipedia). My opinion is that the template should only be added if it serves some encyclopedic value, e.g., if a significant portion of the article in question is actually about a character set. In the case of this article, its presence disrupts the contents of the article, and should be removed. Sławomir Biały (talk) 11:50, 14 May 2010 (UTC)

Topics from 2011-[edit]

English name for set[edit]

Recently I am reading some older sources about the set theory (specially Huntington's The Continuum..., and Jourdain's translation of few Cantor's famous 1895–7 articles (ISBN 978-0486600451)). I see that authors use different names for set {Huntington - "class"; Jourdain - "aggregate"), and I guess that name "set" came into English later, of course most probably from German 'Cantorian' original die Mendge'. Is this true, and is it known when this naming was first used? I have also to figure it out, when name for set was first used, (and by whom) in my native language. --xJaM (talk) 13:55, 21 January 2011 (UTC)

Limits of subsets[edit]

It is usefull to know that A_{n} \uparrow A means A_{n} \supseteq A_{n+1} and A = \bigcup_{n=1}^{\infty}A_{n}. Simirarly A_{n} \downarrow A means A_{n+1} \supseteq A_{n} and A = \bigcap_{n=1}^{\infty}A_{n}. —Preceding unsigned comment added by Boucekv (talkcontribs) 11:24, 5 May 2011 (UTC)

Sets in Statistics[edit]

I have just come across Sets in Statistics while patrolling new pages. A lot of the content looks to be duplicated with this article, but might there be something to be gained from a merge? Best — Mr. Stradivarius 10:40, 21 December 2011 (UTC)

Well-defined sets[edit]

I have deleted some examples that are not aligned with the well-definedness of sets: For example, A = {1, 2, red} does not constitute a set under a well-defined universal set. The union or intersection with such a set is not well-defined, either. — Preceding unsigned comment added by (talk) 12:22, 29 September 2013 (UTC)

It's been a while since you wrote that, but it is a well-defined set in the universe of (a) numbers and colors or (b) strings of ASCII characters. — Arthur Rubin (talk) 09:11, 25 March 2014 (UTC)

Confusing part about uniqueness of sets[edit]

As someone who's just learning set theory, I found this part of the article confusing: "Every element of a set must be unique; no two members may be identical. (A multiset is a generalized concept of a set that relaxes this criterion.) All set operations preserve this property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a sequence or tuple). Combining these two ideas into an example

{6, 11} = {11, 6} = {11, 6, 6, 11}"

The right-hand side of this equation certainly looks to me like something that violates the uniqueness property. If it's not just me that finds that notation confusing, perhaps someone could add an extra line explaining it... — Preceding unsigned comment added by Monsterman222 (talkcontribs) 08:12, 25 March 2014 (UTC)

The elements of a set are unique, so the representation is not unique. I don't knowhow to expalin it, though. Any ideas? — Arthur Rubin (talk) 09:12, 25 March 2014 (UTC)

Bolzano and the term "set"[edit]

Article sez:

The term "set" was coined by Bolzano in his work The Paradoxes of the Infinite.

Is that really accurate? Bolzano was writing in German. Assuming he did in fact coin the term (a point for which I have no independent confirmation but am willing to believe), the term he coined was not "set" but Menge. Right? And the translation from Menge in German to "set" in English is not so canonical that another word might not have been chosen. For example, English wiktionary gives 1. quantity 2. crowd 3. (mathematics) set. I'm not saying Bolzano's contribution here should be overlooked, but I think it may be a little imprecise to say he coined the term "set". --Trovatore (talk) 22:25, 30 April 2014 (UTC)

I have made an edit that addresses this problem. It's certainly possible that a more elegant solution exists. --Trovatore (talk) 22:44, 30 April 2014 (UTC)


3 kinds of set — Preceding unsigned comment added by (talk) 10:13, 5 June 2014 (UTC)

3kinds of set[edit]

3 kinds of set — Preceding unsigned comment added by (talk) 10:15, 5 June 2014 (UTC)


I'm happy with the first conjunct of

the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms

but not the second. There are set theories other than Zermelo–Fraenkel! (talk) 18:14, 26 April 2015 (UTC)

Indeed, axiomatic set theory can be something else than Zermelo-Fraenkel. Correcting. Ladislav Mecir (talk) 23:05, 15 September 2016 (UTC)

Cantor and paradoxes (again)[edit]

This seems to come up over and over, and in fact several of our articles (the worst is probably naive set theory) are seriously problematic on this point.

It is not in fact at all clear that Cantor's informal conception of set, correctly understood, leads to the classical antinomies (the Russell paradox, the Cantor paradox, the Burali-Forti paradox). Gottlob Frege's system was definitely refuted by these paradoxes, but Cantor's was not precisely enough stated to make it clear one way or the other. If you read Contributions to the Foundation of the Theory of Transfinite Numbers, with a bit of historical perspective/bias, it seems very like what would later be called the von Neumann hierarchy, which does not (as far as anyone knows) admit the paradoxes.

Wang Hao took the position that Cantor's theory was not paradoxical. Maria Frápolli, as I understand it, claims to identify a progression in Cantor's thought, with an earlier theory that was paradoxical, and a later one that may not have been.

In any case, while it is not clear that "a gathering together of definite objects of our perception" is paradoxical, what is clear is that it is not a formal mathematical statement, and that by itself probably justifies the claim that it was "inadequate for formal mathematics", as our article states. --Trovatore (talk) 08:22, 16 September 2016 (UTC)

Does it anywhere say here that Cantor's informal conception of set leads to the classical antinomies (the Russell paradox, the Cantor paradox, the Burali-Forti paradox)? Naive set theory seems to be fairly neutral on that point. The mentioned problems definitely arise when unrestricted comprehension is assumed. Perhaps this should be pointed out. But drawing conclusions these days that Cantor's theory way not paradoxical is just as ridiculous as claiming the opposite (since Cantor wasn't precise enough for anyone to draw conclusions). Cantor was a hero, but not a prophet. I think it would be wrong if the words "paradox" and Cantor are disallowed in the same article. YohanN7 (talk) 09:33, 16 September 2016 (UTC)
I think that the statement is fine. Of course, there are plenty of references that claim Cantor's set theory was inconsistent as well, but as Yohan says we don't say anything about consistency in the article. We way that Cantor's definition was inadequate and that contemporary theories instead take "set" as an undefined notion. That claim seems fine to me: in fact, nobody uses Cantor's metaphysical approach anymore (no more "inconsistent multiplicities" etc) and instead "set" is treated as an undefined term. — Carl (CBM · talk) 11:37, 16 September 2016 (UTC)
Sorry, I probably didn't give adequate context. I was explaining my revert of this edit. --Trovatore (talk) 17:17, 16 September 2016 (UTC)

Definition again[edit]

Recently, the lead was changed to say that a set is a well-defined collection of objects. I can accept that the article contains a section called "Definition", but going from there to say that it is well defined is too much. "Set" is an undefined primitive notion. A reference for this (if contested) could be Halmos. YohanN7 (talk) 10:06, 1 November 2016 (UTC)

I see now that the "definition" insists on "well-defined" too. Then it says "objects could be anything". Cantors "definition" of set does not boil down to "a thingie is a set if and only if this and that" in a well-defined manner. YohanN7 (talk) 10:06, 1 November 2016 (UTC)

So first of all the claim in the article does not seem to be that the notion of set is well-defined, but rather that each individual set is well-defined.
But actually that claim is less supportable. It can be argued that "set" is, if not well-defined, then at least well-specified, because of the canonical isomorphism between equal levels of the von Neumann hierarchy, however ∈ is interpreted. The statement as given, unfortunately, suggests that each individual set is determined by a defining formula, which is not true in any non-trivial sense.
I think what Cantor wrote was neither one; rather, the objects of the set were supposedly "well-defined". But I don't think this meant they had definitions. It just meant that the objects were supposed to be particular things. A particular object in a set is not supposed to be one thing for me and a different thing for you, or one thing today and a different thing tomorrow. That's how I've always read it, anyway.
In any case I agree that the current usage of "well-defined" is confusing (and also not supported by Cantor's phrase), and should be changed or removed. --Trovatore (talk) 17:27, 1 November 2016 (UTC)
I think that reading "well-defined collection of objects" as a statement about definability of the set notion is a misunderstading. In my opinion, what Cantor meant by claiming that a set contains "definite" objects is that the set membership is definite, in other words, that the set is not "fuzzy". However, if the formulation causes this kind of misunderstanding, it is probably not optimal. Ladislav Mecir (talk) 20:15, 1 November 2016 (UTC)
Yes, I think you and I are saying more or less the same thing here.
This is the line from the Beiträge, as translated into English by Jourdain:
By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung su einem Ganzen) M of definite and separate objects m of our intuition or our thought.
This is a formulation that actually holds up really well to the present day. I think it would be an excellent basis for the definition in this article, provided we scrupulously avoid inviting the inference that it in itself leads to the paradoxes (though the view that it does lead to them should certainly be discussed).
I suspect that someone in the history of this article, or perhaps some writings that influenced someone writing this article, improperly reformulated "collection of definite objects" to "well-defined collection of objects". We should also make it clear that "definite" objects are not necessarily definable. --Trovatore (talk) 22:41, 1 November 2016 (UTC)
I agree. It is not well-defined in any mathematical sense, but certainly not ill-defined either in any precise mathematical sense. It is just informal. Maybe also note somewhere that common axiomatic set theories make no attempt at a definition of "set" either. YohanN7 (talk) 10:02, 2 November 2016 (UTC)
There are three separate things on the table here: (1) The well-defininedness of the notion of set; (2) the well-definedness of any particular set; (3) the objects being "definite and separate".
I think whoever wrote the language in dispute was conflating (2) and (3), whereas you, Yohan, are conflating (1) and (2). But they are three completely separate things. --Trovatore (talk) 16:01, 2 November 2016 (UTC)
Possibly. But I am actually referring to what I perceive of, in ordinary language, as the idea of set, not any particular set. Thus (1), not (2) if you ask me. I write "Set is an undefined primitive notion" in my first post, where I think I am clear about what I mean. Whether notion is an established concept different from ordinary language idea, I don't know. YohanN7 (talk) 13:30, 9 November 2016 (UTC)
I acknowledge that my last sentence in my first post touches on item (2), but I hope I don't conflate the itemsFace-smile.svg. I am concerned mostly with item (1). YohanN7 (talk) 14:08, 9 November 2016 (UTC)


The text "Cantor's definition turned out to be inadequate for formal mathematics" is certainly true. However, in my opinion, it is also misleading. I think that the formulation "Cantor's definition turned out to be inadequate" is less misleading. Ladislav Mecir (talk) 07:29, 9 November 2016 (UTC)

I don't entirely agree. Cantor's formulation in the Beitraege (at least, as translated by Jourdain) is essentially the modern conception, though less fully explained. To me that makes it problematic to say that it "turned out to be inadequate", especially if the reader assumes that the paradoxes are the reason it "turned out" that way — the paradoxes do not attach to this notion.
It is true of course that those few words of Cantor are not really enough to do much with, without further explanation. At the very least, you need to talk about treating sets as individuals, and collecting them into other sets, and you need to explain the ordinals. If that's what you mean, I don't disagree. But the "turned out" wording is still problematic. Suggest another alternative, maybe? --Trovatore (talk) 21:43, 3 December 2016 (UTC)
'To me that makes it problematic to say that it "turned out to be inadequate"' - For the definition to be adequate, it should be useful as a criterium whether some object is a set. To that you note: "It is true of course that those few words of Cantor are not really enough to do much with". To confirm it, let's try: is the collection of all sets a set? If we consult Cantor's works, we may get Cantor's answer to this question. However, the formulation mentioned as "the Cantor's definition" in the article, i.e., the formulation you call "those few words" is not adequate for that. This is known and discussed in reputable sources. Ladislav Mecir (talk) 23:00, 3 December 2016 (UTC)
By an "aggregate" {Menge) we are to understand any collection into a whole {Zusanivienfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought.
So no, the collection of all sets is not a set. In order to collect them into a whole, they have to be there in the first place. If you collect the ones you already have, then that is a new set, which was therefore not collected.
In any case, I agree that this may not be entirely fully specified by that one sentence. But it is, I think, Cantor's intended conception, and it does avoid the paradoxes.
My main concern here is to avoid giving the impression that the paradoxes are the reason this definition is "inadequate". If we can find a way to clearly avoid that, then it shouldn't be a problem. --Trovatore (talk) 00:07, 4 December 2016 (UTC)
  • the term is Zusammenfassung
  • "So no, the collection of all sets is not a set. In order to collect them into a whole, they have to be there in the first place." - you are trying to input your understanding of Cantor's work to the formulation. However, the formulation mentions "objects of our thought", and what forbids you, me, or anyone else to think about sets? Once we do, they are "objects of our thought", and no additional condition in the "few words" prevents you, me, or anyone else to ask whether their collection is a set, and whether the formulation in question gives us a definite answer. Also note that the existence of a universal set alone does not lead to paradoxes. Ladislav Mecir (talk) 07:48, 4 December 2016 (UTC)