Talk:Set theory/Archive 1

???
"...Russell's paradox...involving "the set of all sets that are not members of themselves." Clearly this set cannot be a member of itself, and hence it must be a member of itself! "

WTF is that supposed to mean ? 201.212.206.205 (talk)

Of course that's a contradiction. That's the point.

$$\lnot \exists\ b \quad [ \forall x \quad (x \in b \iff x \notin x)]$$.--90.13.29.192 (talk) 09:08, 17 September 2008 (UTC)

Russell's paradox is a very important aspect of set theory, and you have to understand it in order to understand why sets are constructed in the way they are, as opposed to the way they were constructed in what they call "naive set theory". Russell's paradox about sets that either are or are not self-membered poses a dilemma for naive set theory, as both options lead to contradictions. This type of proof procedure is standard in logic and mathematics. It's a reductio ad absurdum. Russell is supposed to have written a letter to Frege [the philosopher and logician] to point out the paradox which arose from Frege's Basic Law V. This paradox is taken to show that there cannot be a "universal set", i.e. a set of all sets. That is ultimately why, for example, in the ZF system of set theory, they adopt the axiom of separation as a set-generating rule. In constructing set theory post-Russell's paradox (I don't think he was actually the only person to note this error), the axioms had to make sure they didn't imply that there is a set of all sets! Different systems offered different solutions to this, which is partly why there are different versions of set theory. (The axiom of separation says that sets can only be made of members of other sets; this prevents them all being added together to form a "universal set". That is the most popular view of how to construct sets, and does not give rise to the paradox.) Russell's paradox is a good tool - in proofs, you can often use it to prove that something would constitute a universal set... the proof is perhaps better set out in logic, as to non-logicians this language sounds a little convoluted... But either way, it's very important - this is actually my first contribution to wikipedia, which is why i haven't set up an account! I do not know how to do the proofs on a computer; this proof is very short. Generally, I just found it a bit disturbing that somebody who didn't know about set theory chose to edit the set theory article! ...Russell's paradox may sound strange, but if you've studied or taught any set theory, you'll know that it's very important - and only "WTF" in a kind of, "wow, that's so amazingly elegant - Frege must have been really mad when he got that letter", kinda way. :) Hope that helps. —Preceding unsigned comment added by 86.147.198.20 (talk) 10:49, 26 May 2010 (UTC)

Shatter
The concept of "shatter" or "shattering" is vitally important in the fields of statistical learning theory, empirical processes, and probability theory in general. I have begun to improve the article entitled "shatter", but although it needs more content, it does indeed warrent an article of its own. Moreover, soon, I hope to begin an article on empirical processes, which is not in the Wikepedia yet, but really warrants an entry. I appreciate the editing and improvements made by Trovatore to my new article on shattering (which made many needed corrections to the earlier article, which was a good start, but had some difficulties about sets, subsets, and classes of sets) which I will continue to augment. -- sorry I am new at Wikipedia, but an old time mathematician. Sorry; forgot to sign in this time...will try to remember next time. Thanks again, Trovatore. Would like to discuss more about how we needed to invoke the Axiom of Choice in some of our work on shattering, and did indeed run into Russell's paradox.

Collections of abstract objects
Sets don't necessarily represent collections of abstract objects. A set could comprise the planets, or the Presidents of USA or... anything. Not just abstract!
 * Indeed, but if we define a set as a collection of abstract objects, then theres nothing wrong with it containing elements that aren't abstract, such as planets. Barnjo 09:45, 15 October 2007 (UTC)

list of set theory topics
This list does not yet exist. This is a hint. Michael Hardy 23:20, 14 Jun 2005 (UTC)

Bullets
For such a basic topic, I think bullet mode is too unfriendly. This is not a PowerPoint presentation! Charles Matthews 08:47, 4 October 2005 (UTC)
 * Ok, prompted by Charles, I've made a small start at making this article a bit less "unfriendly". I have integrated some of the ideas from the "PowerPoint" like See also section, but I have, for now left that section untouched. Obviously a lot more needs to be done. But I will wait to see what reaction this evokes in others first ;-) Paul August &#9742; 22:16, 4 October 2005 (UTC)

Self-contradictions in Set Theory
Self-contradiction in set theory arises when the distinction between 2 definitionally distinct concepts is blurred. For examples:
 * Russell’s paradox blurs the definitions of “set” and “element”
 * Cantor’s paradox blurs the definitions of “set” and its “complement-set”
 * Burali-Forti’s paradox blurs the distinction between “set” and “ordinal number”
 * Cantor’s theorem’s bone-of-contention set blurs the distinction between “set” (the “power set”) and “element” (the “subset” as element of the power set and the “element” of the subset)
 * “Completed infinite set” blurs the distinction between “completed” and “infinite” (“incompletable” or “no last element” or “there is always a successor element to every element”)
 * Cantor’s diagonal argument “proving” the “uncountability” of the real numbers blurs the distinction between “set” (which cannot be completed ) and “sequence” (which may be completed); between “interval” or “variable” and real number “point” or “constant”; as well as between the countable enumeration forms x1, x2, x3, … and y1, y2, y3, …, z1, z2, z3, …

Related (as “vacuous truths”) to first-order logic’s material implication self-contradiction is the inherent self-contradiction in set theory --- the empty set is an element of any set’s power set (or a subset of any set); hence, the empty set is also an element of the power set of any given set’s complement-set (or a subset of any set’s complement-set). This obscures the distinction between an element being “included” and “excluded” in a set (or “mutual exclusiveness”) or a “set” and its “complement-set”.
 * Just like the statement calculus and predicate calculus of first-order mathematical logic, the self-contradictions are barred ab initio by agreeing that Aristotle’s 3 “laws of thought” (which are definitionally equivalent) as well as contraposition (which is definitioanlly equivalent to material implication) are to be “first principles” --- that is, over and above all other axioms of any first order theory --- in particular, the first principle of non-contradiction which prohibits the application of a self-contradiction (a logical formula and its negation) at the same time in the same respect.
 * This means that both claims “the empty set is a subset of set S” and “the empty set is a subset of the complement-set of S” could not both be used in one argument.
 * Consider Cantor’s “set of all sets” paradox. If the “set of all sets” exists, then it has no complement-set --- but this means that its complement-set is the empty set which, being a set, is also included in the “set of all sets”.  Hence, the self-contradiction.  Therefore, Cantor’s “set of all sets” is barred ab initio by Aristotle’s first principle of non-contradiction.

Please read my related Wikipedia discussion notes on “Logical conditional”, “Cantor’s diagonal argument”, “Cantor’s theorem”, “Cantor’s first uncountability proof”, “Ackermann’s function”, “Boolean satisfiability problem”, “Entscheidungsproblem”, “Definable number”, and  “Computable number”. (BenCawaling@Yahoo.com [14 December 2005])


 * And speaking of Cantor, I see that he's not mentioned in the article. Should someone maybe add a sentence about him and a link to his page--Georg Cantor?  I'm out of my depth here, so I'll leave it up to you guys. Thanks.--Staple 22:06, 26 April 2006 (UTC)

A few basic questions
Can anyone answer a few basic questions I have about set theory? I am very inexperienced in mathematics in general but in dealing with philosophy, and particularly indeterminacy in philosophy, I have begun to introduce myself to it. Philosophy, like mathematics, deals with the properties of sets, their elements, their boundaries, their unions, et cetera, and attempts to precisely and rigorously define these concepts via logical proof.

I recently began reading the freely-available book Basic Concepts of Mathematics by Elias Zakon. I was impressed at first by Zakon's statement at the beginning of the book that he found rigorous proof lacking in mathematics as it is generally taught to those inexperienced in the field; this has always been a problem for me when I've tried to become familiar with the mathematical concepts others use to describe things like sets. I was thus very surprised by what appears to me to be an extraordinary lack of rigor and, indeed, lack of proof in general of a statement of "fact" in the very first chapter:

Zakon asserts that

"[I]f M is a collection of certain sets A, B, C, ..., then these sets are elements of M, i.e., we have A is an element of M, B is an element of M, C is an element of M, ...; but the single elements of A need not be members of M, and the same applies to single elements of B, C, .... Briefly, from p is an element of A and A is an element of M, it does not follow that p is an element of M. This may be illustrated by the following examples.

Let a “nation” be defined as a certain set of individuals, and let the United Nations(U.N.)be regarded as a certain set of nations. Then single persons are elements of the nations, and the nations are members of U.N., but individuals are not members of U.N. Similarly, the Big Ten consists of ten universities, each university contains thousands of students, but no student is one of the Big Ten."

Can anyone provide me with three sets of actual numbers A, B, and C such that A is an element of B, and B is an element of C, but A is not an element of C? I currently think that this is impossible, and I really need an example of this phenomenon if one exists. This is hindering my progress in learning set theory.

Zakon speaks of "nations" as though they are discrete entities with defined boundaries. This would be necessary if we are to talk about them as sets, in my current understanding. But nations rise and fall and change during the lifetimes of men and women: think of the American Revolution or the fall of the Roman Empire. What makes a "nation" a "nation"? People differ in their answers to this question.

Furthermore, the Big Ten, being universities, consist of teachers, buildings, et cetera, as well as students. The reason that a student at one of the Big Ten is not one of the Big Ten is that a university necessarily contains other things than students; there exists a set whose elements are all elements of the set of all universities but not elements of the set of all students.

These are not "certain" sets, as Zakon says they are, in that their properties and elements are not clearly defined. And if the "fact" of the existence of this phenomenon is so obvious and so necessary to the rest of set theory, then why doesn't he illustrate it with quantifiable sets? Isn't this what mathematics is all about?

I assert that there exist no bounded sets of numbers A, B, and C where A is an element of B and B is an element of C but A is not an element of C. If I am wrong, I'd really like to know. I need to see rigorous proof of this before I can understand anything else in set theory. The concept Zakon is trying to illustrate relates to the definition of what he calls a "family of sets". I have seen a few articles on Wikipedia that talk about things like this; although I can't think of an example at the moment, I'm sure you guys have heard of families of sets. What property could any set C have that allows it to have subsets with elements which are not elements of set C? As I understand it, an element of something is a part of it, it is within it, et cetera: how, then, can set C contain a set B which consists not only of elements of C but of elements not in set C? If the elements of set B are parts of set B, and set B is necessarily part of set C, then any element of set B must necessarily be a part of set C. Again, can anyone give me a real, concrete counterexample that doesn't rely on "intuition"? I thought that in mathematics a thing wasn't supposed to be "more" than the sum of its parts: isn't it central to the very concept of partition that all of the parts of any given thing must necessarily add up exactly to that thing?

Mathematics is supposed to be about rigorous proof, and Zakon claims to be an advocate of such rigour. But I fail to see how things like nations and universities can be considered to be discrete mathematical entities. It seems absurd to me to approach set theory this way. In philosophy, the supposed property to which I have just referred (the one set C would need in the above example to include set B but not all elements of set B) is called a thing in itself. Kant proposed its existence in his Critique of Pure Reason, and Nietzsche extensively disproved its existence. Nietzsche argued that since such a thing cannot in any way be quantified, any two observers cannot be certain that they are both observing it, and that there is thus no compelling reason to suppose its existence to begin with. Any discussion or quantification of it cannot occur, since as soon as it is defined, or bounded, the "thing in itself" would have properties and necessarily not truly be a "thing in itself".

Mathematically, one could define the "thing in itself" as the set that does not share any elements with any other set. It's not even the "empty set", in that it has defining properties which are its elements, such as the ability to be discussed in a language. This leads me to another question: how can there exist an empty set? I can see that it is useful to discuss such a set in defining what exactly an "element" is in general, but in reality there exists no set with absolutely no elements: it would not be a set at all, since a set is a group of elements. The empty set is bounded in that any element of any other set is outside of it. Why do people say that the empty set is an [element] of every other set? Sets are defined by their elements or by the formulae or functions that produce all of their elements. Occam's razor would neatly slice the empty set out of any other set, as far as I can currently tell, if we are using sets to approximate any sort of real phenomena. Even [mere description] of the "empty" set gives it some sort of definition, and thus (supposedly) includes in it certain things (such as not having any elements) and excludes other things (all possible elements of all possible sets) from it, giving it a boundary.

Am I really supposed to believe that such a set has any definite connection whatsoever to any observable phenomenon? I really hope that I'm just misunderstanding this, because if modern mathematical thought centers around concepts that are utterly unquantifiable then the world is a much weirder place than I want it to be.

I also do not understand how a set can be an element of itself. An element of a set is a subset of it, but unless sets are "granular" (I think that's at least close to the right term) there must necessarily be some element in any superset A of any set B that is [not] part of set B: otherwise they would be completely equivalent. Why consider something a subset of itself? How does this make sense? Aren't sets partitioned into their subsets? How can a set A be "partitioned" in such a way that the boundary that defines the "partition" bounds only set A itself and nothing besides? What am I missing here?

I hope that someone can find the time to explain to me exactly what a "family of sets" is, and rigorously. I haven't seen it done yet. This is hindering my progress in learning set theory, and I am having trouble finding a good definition that will allow me to move on to more advanced concepts. Mathematics shouldn't be about "intuition" but about proof. I apologise for using so much space on a discussion page meant for people who (I hope) find my argument silly and can mathematically prove it to be so.

I am also interested in the concepts of similarity and difference: they relate to set theory in that if a set A is similar to set B then it ought to share some elements with set B but not others; but in reality we can continue to find differences between any two things indefinitely. Even in a statement as simple as "1 = 1", there is a definite difference between the two "1"s, no matter how small: if there were no difference at all between them, how could we possibly be considering them as two different "1"s at all? In other words, what elements, exactly, do any two "equivalent" sets actually share? For a set to be completely equivalent to another set, both sets must contain exactly the same elements. There is no magical, mystical concept of "oneness" that allows us to make assumptions about the possibility of true equivalence; quantification is a natural part of human behavior that is necessary to our lives. It was once thought that Euclidean geometry was the only way to describe space, and there are now non-euclidean geometries. But where does this urge to falsely equate things come from? It seems to me that the concept of equality would be better termed increasingly-close approximation. No two separate things can possibly be completely equivalent or they would be a single thing. What possible difference could there be between them?

In other words, if set A is exactly equivalent to set B then they should contain exactly the same elements and neither set should contain any additional elements; but if they are to be discretely named set A and set B, then there surely must be some element not shared between them, or "they" would simply be named set A! Occam's razor would rid us of set B in this example as unneccessary if set A is supposed to approximate some part of reality. What elements could possibly be in a set that is exactly equivalent to a different set?

I realise that set theory is very basic to mathematics, and that is why I must ask these questions. Again, if there is a real, mathematical proof of the thing-in-itself-- or if, more likely, I've mistaken some other concept for the thing-in-itself-- then I apologize for wandering into the wrong "department". But I don't know how else to get this straight than by asking here. Have I stumbled on some of the "contradictions in set theory" mentioned below? Or am I just being really dumb? A little help would go a long way here.

Tastyummy 08:53, 22 August 2006 (UTC)


 * Hi Tastyummy, you ask: Can anyone provide me with three sets of actual numbers A, B, and C such that A is an element of B, and B is an element of C, but A is not an element of C? Consider the sets A = {1, 3}, B = {1, {1, 3}} and C = {{1, {1, 3}}, then B has two elements: the number one, the set whose only elements are the numbers one and three. Since A is "the set whose only elements are the numbers one and three", A is an element of B. Likewise C has one element the set B. Thus B is an element of C. But A is not an element of C, since C's only element B, is not equal to A. Does this make sense? Paul August &#9742; 21:19, 22 August 2006 (UTC)


 * Thanks very much for your reply.
 * I think this makes sense, but let me pose another question to make sure I'm understanding this correctly:
 * Is the set containing no elements different from a superset containing only that set? In other words, if the only "element" of a set A is simply that set which contains no elements, then am I to consider A to have an element after all? I mean, this seems to make sense only if we consider set A { { B, C } } as fundamentally different from { B, C }, right? Is the only thing that distinguishes {no elements} [please exsuse my not knowing how to use the symbol for the empty set] from { { no elements } } and from { { { no elements } } } that we are considering them as supersets and subsets of one another?
 * As I said in my questions above, I am having trouble understanding the concept of the "empty set" altogether. If a set is a group of elements, and the empty "set" contains no elements, then how is it a set at all? I'm sorry if this sounds incredibly stupid. I just don't understand how a set containing nothing is different from a set containing a set containing nothing, et cetera. Surely our consideration of things as sets comes from our ability to group things, right? (Or does it? Am I just reading too much into it?)
 * Zakon says in the beginning of his book that
 * "A set is often described as a collection (“aggregate”,“class”,“totality”,“family”) of objects of any specified kind. However, such descriptions are no definitions, as they merely replace the term “set” by other undefined terms. Thus the term “set” must be accepted as a primitive notion, without definition."
 * If something is altogether undefined, how can we possibly discuss its mathematical properties at all? Isn't definition pretty necessary if we are to "prove" the properties of sets? And if it is an "intuitively-defined group", or something, then ought we not to start at the beginning with what makes a group a group before we move on to other aspects of set theory? Elements of sets always share at least one aspect: that they are the members of that set. But how can we call a set with no elements a set? Where is the property that makes the set containing absolutely no elements different from nothingness in general?
 * I mean, what is not shared between "nothing" and "containership of nothing"? "Containership" itself? What can this possibly mean without any elements being "contained"? Don't we derive our concept of continence from observing actual things which are contained within one another? How can there be "containership" without anything being "contained"? How can this be observed in reality, or is it not even supposed to be? The problem I am having is that if I use a set to approximate a real phenomenon, such as "the set of all atoms of carbon", then it would seem that I could approximate all actual atoms of carbon in set theory equally well by calling that set "the set containing all atoms of carbon and the empty set", and I could also call it "the set containing only the set whose elements are all atoms of carbon and the empty set and another set containing two empty sets", and so forth, without adding anything to or taking anything away from how closely this concept approximates all actual atoms of carbon. Is this where mathematics and science differ, because Occam's Razor seems to suggest this. How can there be such a thing as a set with no elements, if all sets are defined in terms of their elements, and how is a set A { B,C } different from a set D { { B,C } }? Are they only different in that we consider them separately?
 * Ben Cawaling says above that
 * "the empty set is an element of any set’s power set (or a subset of any set); hence, the empty set is also an element of the power set of any given set’s complement-set (or a subset of any set’s complement-set). This obscures the distinction between an element being “included” and “excluded” in a set (or “mutual exclusiveness”) or a “set” and its “complement-set”."
 * I'm having, I think, the same problems here: how can something (the empty set) be an element of both a set and its complement, if the complement of a given set is defined as all elements not in that set? Isn't this a reductio in absurdum? We now have "The empty set is both and element of any given set and not an element of it". How can something be the exact opposite of itself?
 * What, exactly, isn't an element of any superset? I can't think of any real object that I can't categorise in some way or another along with other objects. Even "the entire universe" is an element of "the set of conceptions of reality", et cetera.
 * Thanks very much for your example, though. At least now I see what Zakon is trying to say (I think). I can try to accept that { 1 } is different from { { 1 } } at least in learning the rest of set theory; you've been a great help to me in this and I appreciate it a lot.
 * One more question: is there any category like "philosophy of mathematics" or something to which it might be appropriate to add my article on the indeterminacy of definition? I really think the concept of the "empty set" is strongly related to Kant's noumenon, and I have also cited Zakon's assertion above that sets are not clearly defined (and "indeterminacy in philosophy" refers to such non-definition, or, alternately, indefinability).
 * Tastyummy 22:59, 22 August 2006 (UTC)


 * One more thing: Is this discussion inappropriate for this page? If so, why? I am considering adding a section about set theory to my article on indeterminacy in philosophy, and I just want to be sure that I understand it correctly before doing this. If I add this section, it will cite mathematical theorems, et cetera, at all appropriate points, such as those referred to by Ben Cawalling above on this page. (In other words, I won't make anything up; I'll only quote people who are well-informed on the subject.) If a mathematical theory relies upon undefined concepts at its root, this should be openly discussed.
 * Also, in order to make this discussion officially relevant to changes in the article on set theory, I propose that an article specifically on contradictions in set theory be written and linked to in the main article.

Thanks again, Tastyummy 23:22, 22 August 2006 (UTC)\


 * To answer your last question first, yes such long and general discussion, not directly about writing the article are not really appropriate for this page. The appropriate venue for such questions is Reference desk/Mathematics. However if you want I would be willing to try to answer some of your questions on your talk page. &mdash; Paul August &#9742; 01:56, 23 August 2006 (UTC)


 * The wikipedia talk pages are really only for dicussing article improvements. But I will try to answer some of your queries.  Note, all your misunderstandings stem mainly from your informal understanding of set theory (well, ZF, at any rate).
 * Regarding the empty set: A set being "a group of elements" doesn't not preclude a set being a group of zero elements. Just like zero dollars is a valid amount of money to have, zero items is a perfect valid number of things for a set to contain.  At any rate, the existence of the empty set can be deduced from the other axioms of set theory.
 * Regarding undefined symbols: At this point, I really must insist you look at the ZF. You don't mention it, but in fact there are two undefined symbols in set theory: 'set' and 'is a member of' (which is equivalent to 'is an element of').  The mathematical properties of these symbols are precisely those properties that the axioms, and any theorems deducible from them, allow them to have.  Mainly, what you confuse is what it means to be 'defined' in a theory, and what it means for a theory to have a model.  A 'definition' in set theory is simply a shorthand for some expression that is ultimately reducible to the symbols or first-order logic, 'set', and 'is a member of'.  That these symbols are undefined simply means that you cannot go on further reducing an expression that's purely in  language into simpler symbols.  What it doesn't mean is that there is no meaning to the symbols.  Finding a meaningful connection between a theory and other mathematical objects is the domain of model theory.
 * Regarding powersets: You have confused 'is a member of' (which is equivalent to 'is an element of') with 'is a subset of'. Some examples:
 * 3 is member of {1, 2, 3, 4, 5}
 * {1, 4, 5} is a subset of {1, 2, 3, 4, 5}
 * {1, 4, 5} is not a member of {1, 2, 3, 4, 5}
 * {1, 4, 5} is a member of {1, {1, 4, 5}, 9, 10}.
 * By the definition of a subset, set A is a subset of set B if and only if all memebers of A are also members of B. Which means that the empty set {} is a subset of every set.  And since the powerset of a set S is the set of all it's subsets, the empty set is a member of every powerset.
 * Regarding philosophy of mathematics: Yes, there's an article on mathematical philosophy already. I suppose the concept of 'definability' in maths is fairly interesting, but note that maths, and especially foundations (set theory, mathematical logic, model theory, some other areas) is extremely formal, and so whether ideas like 'definability' and even 'truth' are fleshed out in maths or not doesn't really matter; we can derive almost everything from the formal manipulation of symbols from axioms and rules of inference.
 * Regarding contraditions in set-theory: Depends what you mean by 'contradiction'. There are paradoxes, but these are usually theorems that merely bend our intuition.  There is 'inconsistency', which means that you can prove a proposition 'P' while also being able to prove a proposition 'not P'.  There are no such inconsistent propositions in current, ZF set theory.  The Russell paradox is an inconsistency in naive set theory, but the Russell-paradox set cannot be formed in ZF.
 * Note I am not an expert - my set theory knowledge consists of vaguely recalled bits of undergrad maths glued together by what I can understand from mathworld and wikipedia. But if you've got any other questions, do feel free to continue this on my talk page.  Again, this talk page is for discussing article improvements. Tez 02:36, 23 August 2006 (UTC)


 * Thanks again for your patience. I'll continue to discuss this on user talk pages as you've both requested, and I apologise for taking up so much space here. I did, in fact, realise (but didn't mention) that membership in a set is also, as you put it, an "undefined symbol": this is what I meant by asking what it is that makes us consider things as members of sets or groups at all.
 * I suppose that my questions about definability in general relate more to a discussion of first-order logic than to set theory. Again, sorry for the inconvenience and thanks very much for your time. I will look into ZF set theory and model theory more closely; I greatly appreciate these suggestions.

-Tastyummy (forgot to log in)


 * Regarding one more thing:
 * "We can derive almost everything from the formal manipulation of symbols from axioms and rules of inference."
 * We certainly can, but the axioms and rules of inference of which you speak are not self-evident; they are simply a model and approximation of reality deriving from our continued observation of real phenomena. Nothing is self-evident; hence my inquiry into the origin of even the most basic concepts.
 * I won't waste any more space here. Thanks again to everyone for helping me with this, and for being patient with my long-winded rants.
 * -Tastyummy

Anwers to philosophical objections and a question about adding a new section to the main article
I've managed to answer my own questions about the "existence" of the empty set, etc., mainly via the help of user:tezh, but also via an interesting quotation of Charles Proteus Steinmetz:


 * "Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional."

From a post I made on Tezh's talk page (our discussion consisted of my posting on his page and his posting on mine) after my having figured out something quite obvious mainly because of his patient attention:


 * "If everything must be considered as a set, then there must be some set with no elements."

In other words, in my opinion it is perfectly reasonable to argue against the consideration of all possible phenomena as sets, since to consider every phenomenon as sharing some common attribute with every other phenomenon is to propose the "existence" of a "self-evident" thing in itself; however, it is also perfectly sensible to conclude, from a given set of axioms, that the empty set "exists", or that { { 1 } } is different than { 1 }, since per the axioms of set theory membership in a set is not the same as being a subset of it, even though I still don't fully understand the necessity of this concept.

Tezh further informed me that my assertions about the empty set's ability to approximate any real phenomenon were more relevant to, for example, model theory than to set theory. I agree with this, although I still differ with Tezh over the possibility of the existence of Platonic ideas (or, similarly, of any "self-evident" truths or "immediate certainties" in mathematics or elsewhere), and agree with Steinmetz that mathematical truths are, like all other truths, conditional.

I am considering writing a section on "set theory and philosophy", in which I would explain philosophical objections to various aspects of set theory and expose them as being erroneous for the aforementioned (and other philosophical) reasons. Does anyone object to my doing this? I'd be careful not to draw mathematical conclusions therein; it would be a statement and refutation of the philosophical objection to the seemingly-Platonic or Kantian "implications" of set theory.

Please let me know if this is inappropriate in the article on set theory. I think it would be useful to readers who, like me, struggle with mathematics in general on philosophical grounds in a world where many eminent mathematicians, like Roger Penrose, espouse a Platonic view:


 * "Indeed, I would regard mathematical objectivity as really what mathematical Platonism is all about" - Penrose, in his The Road to Reality: A Complete Guide to the Laws of the Universe

Whether or not "objective truth" is possible, various philosophers have been critical of it for centuries; Nietzsche was especially influential on modern philosophy and, arguably, science in his criticisms of Platonism and Kantianism, and eminent mathematicians like Steinmetz have taken the view that mathematical truth is not "objective" as well.

I wouldn't even go into all of these vagaries if I were to write the proposed section; I'd simply address and refute the specific objections to certain conclusions in set theory which I raised earlier, and as concisely as possible. Again, please let me know whether this is appropriate.

In fact, what I'll probably do is post a draft of the section on this discussion page and then ask whether it's appropriate for the main article. But if there are any objections to my even beginning to do this, by all means, let me know.

Thanks,


 * Tastyummy 20:24, 13 September 2006 (UTC)

Question
Is it possible to ( sensibly) think of sets as identical with their members. Has anyone done so?
 * If by "identical" you mean that (p) a propositon regarding a set S is equivalent(the logical meaning) with (q) a prosition about its members, certainly not(or at least if I understood well your hypothesis)..consider the proposition "S is not empty"(where S={a}) which should be equivalent with "a is not empty" in this view. Anih 00:25, 3 June 2007 (UTC)

Requests
section on "set identities"

Is this "enough" for a set theory article?
The section on objections seems larger than the main body of the text here. Personally I find this to be far less than enough of an article on an important topic such as set theory. I am not sure who stared and manages this page, and which team reviewed it, but I think sets were sold short here. I don't do math pages that often any more... hence the suggestion for someone else to do it.

And the fact that people are asking for a "tutorial" on sets above on this talk page means one thing: The page failed to tell them what they needed to know. I think the article needs serious expansion since it is marked "vital"... poor Georg Cantor .... History2007 (talk) 00:28, 11 March 2008 (UTC)


 * Since no one else volunteered, I will gradually improve this article myself, as my time allows. I will improve it in an incremental way, but will try to keep it in good shape as it progresses. History2007 (talk) 00:30, 15 March 2008 (UTC)


 * The article certainly needs improvement. The question is what it should cover. Right now it is mostly algebra of sets and naive set theory. I would think it should be more of an overall summary article, also pointing to important ideas in axiomatic set theory and descriptive set theory. &mdash; Carl (CBM · talk) 02:37, 15 March 2008 (UTC)
 * Yeah, I agree, except for the usage of axiomatic and naive, to which I've expressed my objections elsewhere. Basically there's an underlying assumption in most of the basic set theory articles on WP that contemporary set theory is about discovering the consequences of axioms, rather than discovering the truth about collections of objects considered as individual objects themselves--essentially, the formalist POV. I disagree sharply with that, but finding a way to name the articles and divide up the material that's more neutral as to foundational philosophy, but also addresses the practicalities involved, is a huge and delicate job, and likely to be contentious. I keep thinking I'll wade into the breach one of these days.
 * I drafted a rough outline of a solution once at talk:naive set theory; I wonder if I might suggest that people interested in improving this article at least look it over. --Trovatore (talk) 02:52, 15 March 2008 (UTC)


 * I think the outline presented there would be a significant improvement over the present article. Something that struck me in the conversation there was a comment about the use of "set theory" to refer to Boolean algebra or Venn diagrams. I also wince when I hear that usage, but it is too established to completely ignore, so some sort of hatnote would be needed to guide readers who are looking for that material. &mdash; Carl (CBM · talk) 03:02, 15 March 2008 (UTC)
 * Thanks, Carl. Any thoughts as to the best way to get started?  The experience with the Boolean algebra articles makes me think there's a risk the whole thing could devolve into a terrible mess if everyone's not on board. --Trovatore (talk) 04:40, 15 March 2008 (UTC)

I looked at that outline and it seems good, but perhaps a bit too heavy to fit into one article. I think if we bite off to much we get nothing, unless yu guys have unlimited time to work on this. I would probably start with a brief intro as to what sets are, then history then selected sections from that outline. Decades ago I decided that the whole debate as to whether Zermelo got it right was pointless for there are so many mathematicians using it that I did not want to spend my life debating it forever. But as one of you said, anything will be an improvement than the current situation that just lists the objections and pushes topus ideas. I think that section is a POV issue anyway for I have not seen large scale demonstrations outside math buildings that demand the fall of set theory and its replacement by the people's republic of topoi. History2007 (talk) 05:21, 15 March 2008 (UTC)

PS: The histoy section (which the outline said needs work) can be based on the following link that may be a good start: And this Stanford entry has a good structure as well History2007 (talk) 05:42, 15 March 2008 (UTC)


 * I think that we can work on Trovatore's outline for this article separately from the question of how to organize our entire collection of articles on set theory. I'll spend some time on it this morning. &mdash; Carl (CBM · talk) 13:44, 15 March 2008 (UTC)


 * I started the expansion, but don't have all of the sections started (most importantly: history and open problems). Much of what I added is still very sketchy, but having a framework in place will give other people a chance to expand it over time. &mdash; Carl (CBM · talk) 15:38, 15 March 2008 (UTC)

Your new infrastructure looks pretty good. But it does need a history section upfront. I always like to know where/how a field started and a history section, however brief, would be very nice. The only issue with the outline is that by the time it s all filled in, there may be too much on one page about set theory and a user who "just wants to find out what set theory is about" will feel lost. So maybe it can be two pages: one front page which is easier to digets and a full page with much more detail. I think when this page is finished, teh answer to my question: "is this enough" will be "yes, it is enough and probably too much". So teh question is: how to make it more digestible for a first reading, yet have the details ina 2nd page. History2007 (talk) 19:11, 15 March 2008 (UTC)


 * What content do you want to see in a "first reading"? Paul August &#9742; 19:22, 15 March 2008 (UTC)

I liked the StanfordEncyclopedia outline, because it was brief and to the point, but we can let the text gather here, as suggested below and then it can be structured based on whatever structures these things. Somehow these take form as they get typed up. History2007 (talk) 03:23, 16 March 2008 (UTC)


 * History2007: I think that some of the present content may get trimmed or moved over time, and the lede would benefit from an eventual rewrite. That may help with your concerns. If the article does become too long at some point, we can worry about splitting it then; getting the information into the article at all is a better first goal I think. I completely agree about the need for a history section, I just don't have any sources at home that would enable me to write it today. &mdash; Carl (CBM · talk) 22:08, 15 March 2008 (UTC)

Ok, let us worry about that separately it it gets too heavy. The source for history was and seems like a good as a start, although it is mostly pre-Zermelo. History2007 (talk) 01:27, 16 March 2008 (UTC)

How much should this page include?
I think this page is gradually taking shape, but I see a trend that will make it hard to read for a simple user. I think we should try to avoid "matehmatical gluttony" so the typical user can get an idea without geting scared on a 1st reading. I deliberately avoided mentioning things like symmetric difference and Cartesian product because they would have been too much to process for a user who just wants a first idea. I think the key here is this: "to those who are used to think in mathematical terms these are trivial concepts, but to a typical user they will be overwhelming".

My suggestion: these sections that refer to a "see main" should be much shorter. For instance, I kept teh Fuzzy set discussion to a simple paragraph and used a simple example just to convey the idea. Had that paragraph discussed linguistic variables, it would have lost the 1st time reader.

So why do we really need Cartesian products upfront if they are in the main article. Alo the use of a lot of bolds makes the text harder to read. But overall, we are making progress I think. History2007 (talk) 20:17, 17 March 2008 (UTC)


 * The article right now is 17k. Lengths up to 30kb, or 6,000 words, are thought to be reasonable in WP:SIZE. We're well short of that here. The history section will need quite a bit of fleshing out; actually most of what I wrote a couple days ago needs to be fleshed out. I just now looked up the AMS subject classification to get ideas about whether what we need to make our coverage complete.
 * Regarding the bold, math articles often put terms in bold when they're defined in the article, although it isn't in MOSMATH. &mdash; Carl (CBM · talk) 20:27, 17 March 2008 (UTC)
 * I think bolding ought to be reserved for the article's title and any terms defined in the article for which this article is their primary article and thus only for terms which should redirect to the article. Paul August &#9742; 20:43, 17 March 2008 (UTC)
 * If that's the convention, I don't mind following it. I don't know exactly what I thought the convention was. &mdash; Carl (CBM · talk) 21:08, 17 March 2008 (UTC)

Let me say it a different way: The American Mathematical Society is for the "mathematically minded" and I was hoping to make the subject more accessible. Believe me that math scares most people, and us mathematicians should remember that. And article length is not a measure of readability as such, for an article of X words on the city of London is easier to read than an article of X words on general relativity. Anyway, it will not make much difference in the end, but my main suggestion will be to avoid indigestion for the typical reader. As is, if someone asks me, where shall I look on the web to find out about set theory, I will have to tell them: "go and read the Stanford Encyclopedia". History2007 (talk) 20:57, 17 March 2008 (UTC)


 * When even mathematicians read a survey article on an area of mathematics they aren't familiar with, they don't expect to understand everything in great detail, but they do expect that the article will expose them to important ideas and give them a sense of what's going on. There is some balance between comprehensiveness and readability, but I don't feel that extra examples of basic set theoretic operations are a large concern compared to the sections on inner models, determinacy, and so on. An unfamiliar reader is going to have to take a leap of faith when reading any article on a field of mathematics.
 * The Stanford article is, to my eye, about half the length that this article should be, and omits many important areas. &mdash; Carl (CBM · talk) 21:08, 17 March 2008 (UTC)

Carl, I think we are not converging towards an agreement. However, given that you are more committed to this topic, I will leave it in your hands. You obviously know the topic well, and if you take it easy on bolds, it may be a good page anyway. Based on my initial suggestion, there is more now than there was before. It is time for me to move on to other interesting topics. Good luck on this article and please try to think of the digestion factor for the users as you edit it. Good luck and best wishes. History2007 (talk) 21:23, 17 March 2008 (UTC)

Things are progressing, but the "naive/axiomatic" dichotomy is still objectionable, and is prominently featured in the lede. This was my biggest complaint all along and the reason I don't really think the change can be done piecemeal. It's good to have a better article here, but this problem needs to be addressed. --Trovatore (talk) 03:01, 18 March 2008 (UTC)


 * Nobody has gotten around to rewriting the lede yet. I think we need to cover the term "naive set theory" in some way, because readers will look for the term. This could be done in the history section, as it is expanded, or could be done lower down. Once there is another section that mentions the term, I'd be glad to rework the lede, which has other issues as well. Please feel free to help... &mdash; Carl (CBM · talk) 03:09, 18 March 2008 (UTC)

Merge of axiomatic set theory
The case for a merge is laid out by Trovatore at Talk:Naive_set_theory. Although the terms naive set theory and axiomatic set theory are important, the division of all of "set theory" between them is misleading to the reader. In the preface to his book, Halmos points outs that his presentation is axiomatic, but justifies the term naive by saying that he is attempting to study the real universe of sets - a practice that many professional set theorists would also say they follow in their research. We can, and should, explain this to our readers. But the current division into two articles (set theory / axiomatic set theory) doesn't serve them, or us, very well. &mdash; Carl (CBM · talk) 14:34, 29 March 2008 (UTC)


 * My 2 cents: We should merge much of the content and redirect from Axiomatic set theory to Set theory. We should move some content to an article called Axiomatic system so as to distinguish from natural deduction systems. Pontiff Greg Bard (talk) 22:19, 29 March 2008 (UTC)


 * Who gets to decide what happens to this article? 71.203.165.35 (talk) 13:43, 6 April 2008 (UTC)


 * There's not a formal process. So far nobody seems to object. If that holds a couple more days, I'll do the merge (not that there is much other content to merge). &mdash; Carl (CBM · talk) 01:21, 7 April 2008 (UTC)

I'm going to redirect axiomatic set theory here. There are a few topics that are covered there, which I think should be covered here, but I didn't think the text from that article would fit here. These include: I hope to think about these and add some text to the article soon. &mdash; Carl (CBM · talk) 18:49, 12 April 2008 (UTC)
 * cardinality (very brief discussion, link to main article)
 * well foundedness (brief discussion), non-well-founded sets

Nice article, now you want to change it?
I had not looked at this page for a while. I must say that over the past several weeks it has seen significant improvements and it looks like a really nice article now - a long way from where it was 2 months ago. Now you guys want to change it? Anyway, do as you wish, but I would let it be for a while until it has settled down, etc. History2007 (talk) 15:32, 15 April 2008 (UTC)


 * I do agree it has come a long way, but I'm not sure what you mean by "you guys want to change it". &mdash; Carl (CBM · talk) 16:56, 15 April 2008 (UTC)

I thought someone was proposing that it should merge with another page, etc. As people on Wall Street say: "mergers always look good at the beginning"... then reality sets in. This is a nice article, no need to rip it apart. Moreover, I do not see why this article has a "start class" attached to it. It is far better than start class in my opinion, so if someone else agrees, please upgrade its status. Thank you. History2007 (talk) 20:51, 15 April 2008 (UTC)

George Boole's contribution
I believe that the history section should give some credit to George Boole, whose 1850's book THE LAWS OF THOUGHT described (apparently for the first time) the ideas of intersections, unions, the null set and the universal set, and how they could be represented by algebraic symbols. Since Boole did not use modern symbols or terminology (He used 0 for the null set, 1 for the universal set, X for intersection and + for union), this is easy to overlook, but the concepts are definitely there, and this work predates all the mathematicians you credited in the history. Since this is the same Boole whose name is immortalized in computer programming with "Boolean variables", you can scarcely argue that he is an obscure mathematician. CharlesTheBold (talk) 04:24, 6 June 2008 (UTC)

Paradox
I'm not sure if this qualifies as a true paradox, but consider this:
 * What is the cardinality of the set of all finite integers? Finite or infinite?

Lucas Brown (talk) 18:25, 25 March 2009 (UTC)


 * No, it is not a paradox. &mdash; Carl (CBM · talk) 01:47, 26 March 2009 (UTC)
 * Note that the class of all sets that are finite or countably infinite, is precisely the class of all "at most countable sets" - as referred to by Rudin in his book on real analysis. The term "finite integer" is imprecise as all integers are essentially "finite" - to expand on this, is to note that a set is finite iff it is not infinite. Equivalently, a set is finite iff it has no proper subset of the same cardinality. Even a set having 1,000,000 elements is considered finite - just as the number 1,000,000 is an integer. If by the set of all "finite integers" you mean a bounded (both below and above) subset of the integers, then the answer is finite. With the most correct of interpretations, "finite integers" refer precisely to the set of all integers, and as such, the answer is infinite. However, in neither case is the statement a "paradox". See paradox for more information - the precise meaning of this term is often confused.

Inclusion of curricula in this article
Should certain curricula be discussed in the opening paragraph of this article? I feel it not relevant to mathematics, and should not be part of this article. If it must, this content should be allocated a separate section lower in the article. -- PS T  05:05, 1 May 2009 (UTC)


 * I think that there is a need to gently clarify, in the lede, that "set theory" is not the same thing as "Venn diagrams", and that the thing that is called "set theory" in elementary school is not the whole story.


 * However, the lede does not particularly discuss curricula; it could be trivially reworded to not use that word. The point of that paragraph is simply to point out that "set theory" is studied at many levels of sophistication. &mdash; Carl (CBM · talk) 18:02, 1 May 2009 (UTC)
 * You are right. I completely forgot about venn diagrams, and how people believe that they fully constitute set theory. However, what about the other articles in mathematics? I guess that people would assume "calculus" to be "calculations" (it is to some degree, but is much deeper than that), or "algebra" to always represent high-school algebra (again, they can't be blamed for this impression but again, mathematics has existed for so long that such an impression only implies mere ignorance), or "mathematics" to be solely about operations on numbers etc... Should we have to include a description of a common misconception that people have about certain mathematical fields (in the corresponding articles)? I realize that people everywhere have the wrong impression about mathematics, but perhaps including an explanation in each article may be too much. Thoughts? -- PS T  22:53, 1 May 2009 (UTC)


 * Well, those are other articles, and can be treated independently. Note that the current wording avoids actually referring to "misconceptions" or "mistakes". Instead, it just explains the scope of set theory in a way that is nonoffensive. Like I said, if you wanted to avoid the word "curriculum" that would be easy to do.


 * Another way of looking at that paragraph is that it is orienting the reader to the topic at hand, which the lede of every article is supposed to do. If you want to reword it some, feel free. &mdash; Carl (CBM · talk) 10:58, 2 May 2009 (UTC)

Computer science
Set theory is a branch of mathematical logic, which is an undisputed branch of computer science. Source:http://wapedia.mobi/en/Outline_of_computer_science#1. —Preceding unsigned comment added by 98.208.55.34 (talk) 07:05, 2 May 2009 (UTC)
 * See my response at Talk:Mathematical logic. Also, the relation "being a branch of" is not transitive. E.g. set theory is a branch of mathematical logic, which is a branch of logic as well as of mathematics, and logic is a branch of philosophy, but mathematical logic is definitely no more a branch of philosophy than it is of computer science.
 * Just because they make you take unions, intersections and power sets of sets in your undergraduate computer science course doesn't mean you are seeing any real set theory there. Sets: yes; set theory: no. --Hans Adler (talk) 07:50, 2 May 2009 (UTC)
 * True. The intended point is that set theory has applications in computer science, but does not constitute a subfield of it. For example, I believe that you will accept mathematics to have applications in physics. Contrary to what it seems, mathematics is certainly not a subfield of physics. In fact, some physicists even accept a large part of physics to be a subfield of mathematics. The main point in your argument, as I interpret it, is simply that since set theory is used in computer science, it is a subfield of computer science. If you are still in favor of the assertion that set theory is a branch of computer science, you must prove it with another argument, as your current argument has found to contain logical flaws by many people. I feel that it is, perhaps, unfortunate that this dispute came to be. I understand that Hans Adler has a strong opinion on this, as do other mathematicians, and myself. However, perhaps it was a bit unnecessary for him to call your edit nonsense, as you were doing it with good faith (presumably). But, it is a valid conclusion that you are false as many people have contradicted your argument although this is not to say that you are certainly false. Based on my limited knowledge of computer science, and my knowledge of mathematics, I can conclude not that you are thoroughly false, but have false logical implications in your argument. I must say that much of computer science would still be non-existent had it not been for the invention of set theory. Therefore, it is reasonable to conclude that set theory forms an important foundation for computer science. However, it is not reasonable to conclude, now, that it is a subfield of computer science. If this is the essence of your argument, it is important to understand that arguments, on Wikipedia, must be supported with correct logical reasoning and/or the provision of evidence. I do not discourage you for arguing further, but if you choose to do so, my advice may be of use to you. This, however, depends on how you use it. -- PS T  10:12, 2 May 2009 (UTC)


 * Set theory is not, by the ordinary way the terms are used, a "branch" of computer science. Set theory is, of course, useful for computer science, like it is useful for economics. It's a similar relationship as the physics&math; one; math is not a subfield of physics, and physics is not a subfield of math, even though the two influence each other. &mdash; Carl (CBM · talk) 11:04, 2 May 2009 (UTC)

I have provided sources, you have not. Until you provide sources, your edits will be reverted. —Preceding unsigned comment added by 98.208.55.34 (talk) 20:46, 2 May 2009 (UTC)
 * I think you misunderstand what are recognized as "sources" in Wikipedia. Wikipedia itself is not a reliable source, and certainly mirrors of Wikipedia are not reliable sources.  Wikipedia is a tertiary source; sources for Wikipedia should be mostly secondary sources, though in some cases primary sources are acceptable.  It's reasonable to add a tertiary source in addition to secondary sources, if the tertiary source is easier to access; also, it's admittedly better than nothing, as a stopgap until a reliable secondary source can be found.
 * But to claim it as some kind of trump card in a dispute &mdash; no, that just doesn't fly. --Trovatore (talk) 20:55, 2 May 2009 (UTC)
 * As I have already pointed out, the essence of your argument seems to be, "without set theory a large part of computer science would not exist and therefore set theory is a subfield of computer science". The argument against this is that the purpose of computer science is not to do set theory. Essentially, a large part of mathematics is to do set theory. Of course, it is much deeper than that, but it is fair to assume that set theory is crucial to mathematics. Without set theory, mathematics would not even exist. On the other hand, I am certain that computer science would exist independent of set theory or not. Furthermore, set theory generalizes to fields such as axiomatic set theory. Essentially, therefore, set theory is built on a set (!) of axioms and this is in some sense unique to mathematics. See axiom. However, no subfield of computer science, in my knowledge, is completely axiomatic. Therefore, set theory cannot be a subfield of computer science. Unless you point out any logical flaws in this proof, I am not convinced. Before providing sources, I feel it necessary that you provide a logical argument supporting your assertions - the logical implications of which will be supported by sources. This enhances your argument. Currently, however, I am unsure of what your argument is, in favor of your statement. -- PS T  00:51, 3 May 2009 (UTC)
 * The other point to note is that, as it seems, your assertion is so out of the ordinary, that it is impossible to find sources contradicting it. Clearly, therefore, people find it too nonsense an assertion, also too trivially incorrect an assertion, that no proof is necessary to demonstrate its falseness. This is one possibility for why there are no sources of the nature which you demand. As I have mentioned previously, sources are usually secondary to a well-structured logical argument. -- PS T  00:56, 3 May 2009 (UTC)

A quote, if we really need a source:
 * Why learn Set Theory? Set Theory is an important language and tool for reasoning. It’s a basis for Mathematics—pretty much all Mathematics can be formalised in Set Theory. Why is Set Theory important for Computer Science? It’s a useful tool for formalising and reasoning about computation and the objects of computation. Set Theory is indivisible from Logic where Computer Science has its roots.

From Glyn Winskel's course Set Theory for Computer Science. The quote clearly conceives of computer science as a discipline separate from logic, and so separate from set theory. One can furnish many similar citations. &mdash; Charles Stewart (talk) 11:05, 3 May 2009 (UTC)

History section
I just did a few little rewrites of the History section, to make it clear and encyclopaedic in tone. If anyone disagrees with a change, please mention it below for discussion.

In particular, I deleted entirely "It was later realized that these paradoxes are not merely set theoretic, and that in logic the sentence "this sentence is false" gives rise to a similar problem, for if the sentence is true, it must be false. Kurt Gödel used this fact in the 1931 proof of his celebrated incompleteness theorem."

This is false, misleading and irrelevant. "This sentence is false" is a semantic paradox, as opposed to the class of logical or set theoretical paradoxes; this distinction between the paradoxes is widely accepted today. Godel did not use this fact, he constructed a sentence which could be viewed as expressing a similar idea. Finally, I don't see what relevance the liar paradox or Godel's theorems have in a discussion of the history of set theory, which they did not (directly, at least) influence; these belong better in the articles on paradoxes, which an interested reader can find more about by following the link to Russell's Paradox.

I'm still not happy with the overall tone of the section, so will probably try a more major rewrite soon; if you're interested, please let me know and we can discuss any changes here. --Joth (talk) 07:49, 10 June 2009 (UTC)

Let me suggest that you should revise the history section, and in so doing you might consult this web page: http://plato.stanford.edu/entries/settheory-early/ In particular, the idea that Cantor was the single creator has been questioned. My book in the references (# Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7) does not quite agree with the views in the older Johnson, Philip, 1972. A History of Set Theory.

What is the source to state that Zeno of Elea was a mathematician?--Gonzalcg (talk) 02:41, 30 December 2011 (UTC)

Finite sets
There are no contradictions in the theory of finite sets. —Preceding unsigned comment added by 86.137.170.8 (talk) 09:07, 23 July 2009 (UTC)
 * What is the context of this remark? -- PS T  09:22, 25 July 2009 (UTC)

Bishop quote
I am going to remove the "quote" from Bishop. First, it was added in January 2004 ago by an IP editor with 5 total contributions, who stopped editing in January 2004. So we are unlikely to get a response from that person about the source of the quote. Moreover, the original text here did not even have quotation marks, they were added by a different editor later.

Second, I have read a decent amount of Bishop's writing. I think that the "quote" here is actually a paraphrase of Bishop's comments at the beginning of Foundations of Constructive Analysis. But Bishop is not talking about set theory there, he is talking about classical mathematics in general. Bishop actually says, "If God has mathematics of his own that needs to be done, let him do it himself". But the context of that paragraph is the positive integers, and Kronecker's claim they were invented by God. Bishop is not talking about set theory in that quote.

Of course one cannot decisively "prove" a quote is not real, but this one smells somewhat fishy to me, so I am going to remove it. If anyone can actually find a quote from Bishop that mentions both God and set theory, then by all means add that back to the article. &mdash; Carl (CBM · talk) 15:02, 11 November 2009 (UTC)


 * I agree, I think it was a corruption of the quote you mention. --99.245.206.188 (talk) 00:58, 12 November 2009 (UTC)
 * If you feel that the quote should be removed, then I concur with you. I have absolutely no experience with regards to the quote. -- PS T  07:02, 12 November 2009 (UTC)

Venn Diagrams
The introductory portion of the article says:

"Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects."

Venn diagrams are NOT used in primary school. The diagrams used in primary school predate Venn diagrams and are often more accurately referred to as Euler Diagrams. Euler used these diagrams in published work, such as "Letters to a German Princess", although they are known to also predate him. Venn diagrams are relatively advanced conceptually and are best understood on the basis of prior knowledge of Euler Diagrams. I believe that Venn Diagrams are simply too subtle and sophisticated for use at the primary school level.

Mathematics texts and articles nearly always misuse the term "Venn diagrams" in this way. Please see the last part of this comment marked with * for a more detailed explanation and links to related articles.

I have tried to correct the Wikipedia article on Venn diagrams, but those changes have been repeatedly undone.

In the present article, I would like to make a change, but the error occurs in the introductory section which has no edit option.

Can somebody explain how to edit that section?


 * Here are some articles on Venn Diagrams. the page at

http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html

is a general definition

The following comment is key: "Note that some authors refer to diagrams with fewer than 2^n regions as Venn diagrams, but they are more properly termed Euler diagrams, after the mathematician Leonard Euler."

In a Venn diagram all intersections are shown as non empty but are indicated as empty by shading, while in Euler diagrams, emptiness of an intersection is indicated by disjoint interiors (which would disqualify a diagram from being a Venn diagram).

Most modern math books make the same mistake and use Euler Diagrams while calling them Venn Diagrams. As far as I can tell, most modern (western) philosophy books do not make this mistake. They usually use Venn Diagrams and correctly call them Venn Diagrams.

You can find more discussion of Venn Diagrams and the distinction between them and Euler Diagrams in the following sources.

The lower part of the page at

http://www.combinatorics.org/Surveys/ds5/VennOtherEJC.html#euler

shows the comparison correctly. In this article, the notation "AB" means "A intersect B".

This is further explained in the following:

http://www.cut-the-knot.org/LewisCarroll/VennDiagrams.shtml

and

http://www.cut-the-knot.org/LewisCarroll/devlin.shtml

which show how the Venn Diagram is used to solve a syllogism.

The following is by one of the people most to blame for the confusion about what Venn Diagrams really are --- William Dunham. This webpage incorporates the challenge from the author of the Survey of which the pages at the first two urls above are parts. It also includes Dunham's response to the challenge to his misinformation, showing only a cavalier disregard for the educational damage he has inflicted on millions. I definitely recommend reading this to see an example of how the misunderstanding of Venn Diagrams became so widespread.

http://www.cut-the-knot.org/LewisCarroll/dunham.shtml

Dagme (talk) 20:36, 16 November 2009 (UTC)


 * We are not here to remedy historical injustices. If the diagrams in question are standardly called Venn diagrams (and I think it's clear that they are) then we will call them Venn diagrams, whether Venn had anything to do with them or not. --Trovatore (talk) 20:43, 16 November 2009 (UTC)


 * I agree with Trovatore. Also the term "Venn diagram" is widely used in the curriculum materials. This document has a 4th-grade-level objective about them. This is a lesson plan on "Introducing the Venn Diagram in the Kindergarten Classroom". If even the educators call them "Venn diagrams", I think we are safe using that term here to refer to what the educators are doing. Also, it does not seem that Venn diagrams are too complicated for primary school students. &mdash; Carl (CBM · talk) 21:01, 16 November 2009 (UTC)

I am going to slaughter this with facts. Here is Venn himself, and he treats the history with great respect. In fact, his diagrams show all 2n "compartments" in a manner similar to a Karnaugh map, and unlike the diagrams we find in most books. He appears to gain the credit because he finally treats the matter of diagrams completely for the first time. Here is what he has to say:

JOHN VENN, 1881, SYMBOLIC LOGIC, MACMILLAN AND CO.

CHAPTER V. DIAGRAMMATIC REPRESENTATION.


 * THE majority of modern logical treatises make at any rate occasional appeal to diagrammatic aid, in order to give sensible illustration of the relations of terms and propositions to one another. With one such scheme, namely that which is commonly known as the Eulerian, every logical reader will have made some acquaintance, since a decided majority of the familiar treatises make more or less frequent use of it1. Such a prevalent use as this clearly makes it desirable to understand what exactly this particular scheme undertakes to do and whether or not it performs its work satisfactorily.


 * As regards the inapplicability of this scheme for the purposes of a really general Logic something was said by implication in the first chapter, for it was there pointed


 * 1 Until I came to look somewhat closely into the matter I had no idea how prevalent such an appeal as this had become. Thus of the first sixty logical treatises, published during the last century or so, which were oonsuIted for this purpose:-somewhat at random, as they happened to be most accessible :-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Soheme.


 * out how very ·special and remote from common usage is the system of propositions for which alone it is an adequate representation. To my thinking it fits in but badly even with the four propositions of the common Logic to which it is usually applied, but to see how very ineffective it is to meet the requirements of a generalized or symbolic Logic it will be well to spend a few minutes in calling the reader's attention to what these requirements are.


 * At the basis of our Symbolic Logic, however represented, whether by words by letters or by diagrams, we shall always find the same state of things. What we ultimately have to do is to break up the entire field into a definite number of classes or compartments which are mutually exclusive and collectively exhaustive. The nature of this process of subdivision will have to be more fully explained in a future chapter, so that it will suffice to remark here that nothing more is demanded than a generalization of a very familiar logical process, viz. that of dichotomy. [etc] (p. 100-101)

Footnote on p. 106:


 * 1 A brief historic sketch is given in the concluding chapter of some previous attempts, before and after Euler, to carry out the geometric notation of propositions. I tried at first, as others have done, to illustrate the generalized processes of the Symbolic Logic by aid of ilia familiar methods, but soon found that these were quite unsuitable for the purpose. Though the method here described may be said to be founded on Boole's system of Logic I may remark that it is not in any way directly derived from him. He does not make employment of diagrams himself, nor does he give any suggestions for their. introduction

Footnote on p. 112:


 * 1 Other logicians (e. g. Schroeder, Operations kreis, p. 10; Macfarlane, Algebra of Logic, p. 63) have made use of shaded diagrams, but simply to direct attention to the compartments under consideration, and not, as here, with the view of expressing propositions.

In his Chapter XX HISTORIC NOTES pp. 405-438 Venn gives an extensive survey of the efforts to "graph" logical concepts. We get to the nitty-gritty here:


 * We now come to Euler's well-known circles which were first described in his Lettres a une Princesse d'Allemagne (Letters 102-105). The weak point about these consists in the fact that they only illustrate ill strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions1. This defect must have been noticed from the first in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well :
 * [intersection-diagram here of both] Some A IS B, Some A. is not B, }


 * 1See the discussion on this point in ch. I, p. 5.


 * for the real relation thus exhibited by the figure is of course "some (only) A is some (only) B", and this quantified proposition has no place in the ordinary scheme. (p. 424)

Wvbailey (talk) 21:46, 16 November 2009 (UTC)


 * Will, you know, bully for Venn and all that, but this material is not terribly relevant to the question at hand. We follow the usage in the field, not what Venn said in 1881.  You need secondary sources, not primary, and more recent ones. --Trovatore (talk) 21:56, 16 November 2009 (UTC)

Response from Dagme:

It is difficult from the format to keep track of who said what, but I will do the best I can in this response. I will start from the end of my comment and work down.

"If the diagrams in question are standardly called Venn diagrams (and I think it's clear that they are)..."

This premise is incorrect. They are called Venn Diagrams by mathematicians who don't know better. They are not called Venn Diagrams by mathematicians who know what Venn Diagrams are. Venn Diagrams are a topic for ongoing mathematical study. Use of the term "Venn Diagram" for something else has led to confusion in the past and continued use of the term "Venn Diagram" for something else will lead to further confusion in the future.

The earlier diagrams are also not called Venn Diagrams by philosophers, who are taught correctly what Venn Diagrams actually are.

Using the term incorrectly serves no purpose and does a disservice to those who are misled by such incorrect usage.

"... then we will call them Venn Diagrams, whether Venn had anything to do with them or not."

The issue is not whether Venn had anything to do with them. The issue has to do with the fact that the term "Venn Diagram" refers to something else, namely the more sophisticated diagrams that were invented by Venn (to remedy the shortcomings of the earlier diagram system).

"Also the term "Venn diagram" is widely used in the curriculum materials."

This is exactly my point. The purpose of Wikipedia is to provide accurate information, not to perpetuate the continued dissemination of MISinformation.

"even the educators call them "Venn diagrams" "

The fact that many people commit an error is not justification for committing the same error again.

"it does not seem that Venn diagrams are too complicated for primary school students"

If you believe this, try teaching Venn Diagrams to primary students who are not already familiar with the simpler earlier diagram system.

"this material is not terribly relevant to the question at hand"

Of course it's relevant. The purpose of Wikipedia is to provide information, not MISinformation.

"We follow the usage in the field"

It is the people who continue incorrect usage who are responsible for the continuation of incorrect usage. Those in the field of mathematics who know better, do not use this incorrect terminology. Those in the field of philosophy nearly all know better and also do not use the incorrect terminology. Please remember also that Set Theory is a discipline in philosophy as well as in mathematics.

"You need secondary sources, not primary, and more recent ones."

I don't know the rationale behind this, but the fact is that accurate secondary sources use correct terminology and inaccurate secondary sources use incorrect terminology.

Finally, my original question: How can I edit the introductory section?

Dagme (talk) 02:15, 17 November 2009 (UTC)

From this survey and the research I conclude that Dagme is correct. This comes as a surprise to me; without the time spent to research this, I would have reflexively agreed with Trovatore and CBM. In fact I set out to prove Dagme full of BS but based on the evidence I now agree with Dagme.

Of course lazy minds are free to inherit and then sustain the canards and gaffes of other lazy minds, but some of us don't like that sort of thing. The above quotation from Jevons 'is relevant if we are to understand exactly what a Venn diagram really is, as opposed to what lazy writers of secondary and tertiary sources (wikipedians) merely opine about hat they think'' it is. Ditto for an Euler diagram. Here is some more "secondary" literature comments. You will quickly see that (if you actually read the above and the below) that an Euler diagram and a Venn diagram are NOT the same thing, but they have been conflated over the years. They are technically NOT the same, and at least one modern author (Bender and Williamson) uses the notion correctly: a true Venn diagram always shows all the 2n regions of n variables; thus they are very difficult to draw with 5 and more variables, and they are not very good for demonstrating set-theortic concepts in a simple way; Venn diagrams are more suited to analyzing propositional logic (and nowadays there are better ways such as Karnaugh maps):

(1) LOUIS COUTURAT, 1914, THE ALGEBRA OF LOGIC, AUTHORIZED ENGLISH TRANSLATION BY LYDIA GILUNGHAM ROBINSON, B. A. WITH A PREFACE BY PHILIP E. B. JOURDAIN. M. A., THE OPEN COURT PUBUSHING COMPANY, CHICAGO AND LONDON


 * "48. The Geometrical Diagrams of Venn.-PORETSKY'S method may be looked upon as the perfection of the methods of STANLEY JEVONS and VENN. Conversely, it finds in them a geometrical and mechanical illustration, for VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only strike out (by shading) those which are made to vanish by the data of the problem. For instance, the universe of three terms a, b, c, represented by the unbounded plane, is divided by three simple closed contours into eight regions which represent the eight constituents (Fig. I). (p. 73-74).

His figure I shows the existence of a region a'b'c' that lies outside the three intersecting circles. Strangely, Couturat concludes that the Poretsky/Jevons method is not very useful (cf p. 74) but goes on to consider tabular methods and then calculate that there are (2)^(2^n) possible functions, given n variables. (pp. 73-80). Euler diagrams are not mentioned.

(3) Patrick Suppes, 1957, Introduction to Logic, Dover Publications, Inc. Mineola, NY, ISBN: 0-486-40687-3.:
 * Footnote p. 195: "† The basic idea of using circles in this way was due to the eighteenth-century Swiss mathematician Euler. Some of the refinements to be explained below are due to the ninetheeth-century British logician Venn. The diagrams are called Euler diagrams, or Venn diagrams".

Suppes progresses from the simple Euler diagrams (drawn inside a rectangle representing a universe). He then proceeds to shade the diagrams, and then draw three overlapping circles in the manner of Venn, together with shading also in the manner of Venn. He marks some of the regions with X's and links them.(cf pages 195-201). He ends with:


 * "Venn diagrams may be used to represent any argument which does not involve more than three sets. Moreover, by a careful use of ellipses in place of circles relations among four sets can be represented diagrammatically, but relations among five or more sets can ofthen not be represented by any simple diagrammatic device." (p. 201).

(4) From R. L. Goodstein, first published 1963, republished 2007, Boolean Algebra, Dover Publications, Inc., Minola, NY, ISBN: 0-486-45894-6.:
 * "1.99 Venn diagrams. The representation of classes by overlapping circles, known as Venn diagrams (or Euler diagrams) is a valuable visual aid in the understanding of class relations." (p. 17)


 * Goodstein draws the diagrams as true Euler diagrams, i.e. without a universe of discourse box that surrounds the circles.

(5)Edward A. Bender and S. Gill Williamson, 2005, A Short Course in Discrete Mathematics, Dover Publications, Inc. Mineola, NY, ISBN: 0-486-43946-1.:


 * Bender and Williamson draw a true Venn diagram, i.e. in their example three ovals with all 7 regions shown, inside a square box they call U for the universal set (i.e. the 8th region). (cf p. 83). They then demonstrate the notion of a truth table from this drawing. (cf p. 84-85)

In a footnote at least the article should point out that Venn diagrams and Euler diagrams are not technically the same thing, but that simple Euler diagrams are often called Venn diagrams. Bill Wvbailey (talk) 02:56, 17 November 2009 (UTC)

You don't have to go to great lengths to find secondary sources which give a correct account of Venn Diagrams. As I said, authors whose background is in philosophy routinely describe Venn Diagrams correctly. Just look at the Stanford Encyclopedia of Philosophy article at http://plato.stanford.edu/entries/diagrams/

Also, just about any modern philosophy text on introductory logic and critical thinking will have a correct explanation.

Additionally, you can check the Oxford Dictionary of Philosophy and, more significantly, the definitive and encyclopedic work by Kneale and Kneale: "The Development of Logic" (1962 and 1984) which says of Venn on page 420:

"His diagrams differ from those of Euler in that he first represents all possible combinations by distinct areas and then indicates by marks within the various areas which combinations must be null and which not null for the holding of a given proposition."

A succinct and accurate description indeed.

My guess is that those contributors who insisted that I was misguided in my initial criticism did not even bother to look at the sources I provided there.Those sources are all worth looking at, as well as what I have given here.

Dagme (talk) 05:29, 17 November 2009 (UTC)


 * We are discussing this sentence:
 * "Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects.""
 * I have pointed out several sources where Venn diagrams are used in primary school and it is easy to find many more by searching for lesson plans on Venn diagrams. On the other hand, no actual diagrams are used in this article at all. So any issue about a distinction between Venn diagrams and Euler diagrams seems to be irrelevant here; there is plenty of evidence to back up the claim that "Venn diagrams" are used in primary schools, which is all that the article claims.


 * Independent of that, I don't see any benefit to adding a long explanation of the difference between Venn diagrams and Euler diagrams to this article. The term "Venn diagram" is usually used to refer to all such diagrams in mathematics, as the secondary sources listed by Wvbailey suggest. For example:
 * "The diagrams are called Euler diagrams, or Venn diagrams"
 * "The representation of classes by overlapping circles, known as Venn diagrams (or Euler diagrams) is a valuable visual aid in the understanding of class relations."
 * These clearly suggest that the authors feel that either of the two terms can be used to refer to the a single diagram. &mdash; Carl (CBM · talk) 14:52, 17 November 2009 (UTC)


 * CBM and I crossed efforts here. As you will observe, I've made the change, but added a footnote. I suggest all should read the wiki articles Euler diagram and Venn diagrams. We could reverse the sense of the thing -- use Venn diagram in the article and change the footnote. But the link Venn diagram will be very confusing, and simply wrong. Bill Wvbailey (talk) 15:10, 17 November 2009 (UTC)
 * Why "wrong"? Lesson plans such as this really do use Venn diagrams, even in the stricter sense of not including a box for the universal set. &mdash; Carl (CBM · talk) 15:27, 17 November 2009 (UTC)


 * Carl, I looked at the link. Sigh. I do understand your point. Even my old college text Kemeny (!) et. al. ca 1958 calls them "Venn diagrams", that's what I always called them too, until a day ago. (As to "wrong": To exhibit a technically correct Venn (as opposed to Euler) diagram the authors of your link need the box, or they need to label the outer area e.g. ~a & ~b or a'b', as did Couturat; thus they will have all four areas identified. Where they will run into difficulties is when they draw three circles all of which do not intersect, e.g. when they are drawing one circle inside another e.g. to show that humans are a subset of mammals.) This might be better
 * "Elementary facts about sets and set membership can be introduced in primary school, along with Euler diagrams (often called Venn diagrams), to study collections of commonplace physical objects."
 * I do think that somehow the fact that Venn diagrams and Euler diagrams are often grouped into the class "Venn diagrams" needs to be pointed out. (Actually, that's kind of an interesting set-theoretic example, isn't it?) Bill Wvbailey (talk) 15:54, 17 November 2009 (UTC)
 * I don't think that this is the right article in which to point that out. The sentence in question is already a side topic, that elementary set theory is taught in the guise of Venn diagrams in primary school. To then go into the difference between Venn and Euler diagrams would be a side topic on a side topic. The article on Venn diagrams does already try to explain the difference, and it is much more relevant in that context. &mdash; Carl (CBM · talk) 17:44, 17 November 2009 (UTC)


 * My two cents. The technical distinction between Venn and Euler diagrams, or the fact that people may have the two notions confused is irrelevant to this article, and in fact, the mention of diagrams at all is not strictly necessary.  However the virtue of including a reference to "Venn diagrams" is that many people will recognize the name and the concept and this will help provide important context. But this virtue will be greatly diminished if we refer to them as "Euler diagrams", a name that I'm confident most people will  not have heard of. If we insist that using "Venn diagrams", without explanation,  is too inaccurate, and that providing an explanation is too much of a diversion,  then it might be best to drop the mention altogether. Paul August &#9742; 18:47, 17 November 2009 (UTC)


 * It would be an unfortunate turn of events if we don't say "Euler" because that term is so unknown, but don't use "Venn" because of some belief that "Euler" is actually the correct term. If (as I believe is true) the usual term in mathematics is "Venn diagram", I would rather just say that and be done with it. Compare the search results for nctm "venn diagram" (10,200 hits) and nctm "euler diagram"  (48 hits). &mdash; Carl (CBM · talk) 20:01, 17 November 2009 (UTC)


 * (ec) I am not convinced that it is "inaccurate" to call them Venn diagrams. Oh, probably historically inaccurate, sure.  But I don't care, and neither should anyone else.  Historical accuracy in naming does not matter even slightly for our purposes.
 * The references to modern workers who do make the distinction, on the other hand, is relevant to us. But when different workers in the field use terminology differently, we do not decide which of them are "correct".  And certainly not on a historical basis. --Trovatore (talk) 20:06, 17 November 2009 (UTC)


 * I am in essential agreement with what Carl and Trovatore have written immediately above. But to Carl, although admittedly it would be "unfortunate" if we leave out mention of diagrams altogether, better that than a bald mention of "Euler diagrams". Here are my preferences:
 * "Venn" but with some qualification (probably best in a footnote).
 * "Venn" unqualified.
 * Drop the mention altogether.
 * "Euler" with some qualification (probably best in a footnote).
 * "Euler" unqualified. (What we have now.)


 * Paul August &#9742; 21:30, 17 November 2009 (UTC)


 * I don't like the footnote because this is in the lede of the article, but is an extremely tangential point. How about this:
 * "Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams and Euler diagrams, to study collections of commonplace physical objects."
 * That includes "Euler" but avoids a footnote. &mdash; Carl (CBM · talk) 22:25, 17 November 2009 (UTC)
 * That's better than what we now have. Better yet is "along with Venn and Euler diagrams", which flows better (though the linking is a bit problematic, as some might expect that the link for "Venn" will end up at John Venn, but I don't think that should bother us much).   Paul August &#9742; 22:52, 17 November 2009 (UTC)

Isn't it a bit ridiculous that we have separate articles for Venn diagram and Euler diagram, anyway? Perhaps the two should be merged, under the more common name Venn diagram, or perhaps Venn and Euler diagrams. A merged article could define Euler diagrams, introduce Venn diagram as a synonym, and say that Venn's original definition was more restrictive and how. Hans Adler 23:34, 17 November 2009 (UTC)


 * I'd keep them separate. (But I agree that some explanation of differences and usages would be appropriate ...). My sense of the (true) Venn diagram is that it is now a backwater of mainly historical importance, but it is an important one on the way to the development of the Karnaugh map which is still a seriously-useful tool. On the other hand the "Euler diagram" is still useful for teaching purposes -- a visualization aid for solving thought-problems (see my explanatory examples at Talk:Law of excluded middle). I'll now verify and demonstrate this assertion about (true) Venn diagrams:
 * I pulled open one of my switching-theory books -- Hill and Peterson 1968, 1974 Introduction to Switching Theory and Logical Design, John Wiley & Sons NY, ISBN 0-71-39882-9. This is an engineer's book. In chapter "Boolean Algebra" they present sections 4.5ff "Set Theory as an Example of Boolean Algebra" and in it they present the (true) Venn diagram with shading and all. They actually use Venn diagrams to solve example switching-circuit problems, but end up with this statement:
 * "For more than three variables, the bsic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." (p. 64)
 * In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:
 * "The Karnaugh map1 [1Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram." pp. 103-104
 * And that is the end of the Venn diagram in this book of 596 pages. I consulted three more engineering texts. Two of them -- (i) McClusky 1965 Introduction to the Theory of Switching Circuits, McGraw-Hill Book Company, NY, no ISBN; and (ii) Wickes 1968 Logic Design with Integrated Circuits, John Wiley and Sons, Inc, NY, no ISBN -- present (true) Venn diagrams the same way as did Hill and Peterson, that is as a feature of Boolean algebra, and perhaps as Wicks calls them "Minimization Aids". As an engineer I never use Venn diagrams, I always use Karnaugh maps (and have 1000's if not 10,000's of times so far). But, to explain simple set-theoretic concepts to myself I do use "Euler diagrams" (both with and without a box around them, which until 2 days ago I thought were Venn diagrams).
 * To sum up, I'd keep the articles split mainly because of their historical significance. Otherwise we could use similar reasoning to fold Euler diagram into Venn diagram and both into Karnaugh map and then Karnaugh map into Hypercube. I think alot of this fussing has to do with didactic and historical precision, which as a historian is something I am very sensitive to. Bill Wvbailey (talk) 15:52, 18 November 2009 (UTC)

Interestingly, there is a Wikipedia article on Karnaugh maps at http://en.wikipedia.org/wiki/Karnaugh_map The Stanford Encyclopedia of Philosophy does not have a stand alone article on Karnaugh maps but discusses them under some other headings. Dagme (talk) 18:40, 18 November 2009 (UTC)

"Isn't it a bit ridiculous that we have separate articles for Venn diagram and Euler diagram, anyway? Perhaps the two should be merged" Yes, this is a good point and a good suggestion. It is what was done in the Stanford Encyclopedia of Philosophy. Since Wikipedia has a feature that redirects searches, both "Venn Diagram" and "Euler Diagram" could be redirected to a single article. The historical development could also be addressed there. The name of the article could be "Euler and Venn Diagrams". It could also be "Set Theoretic Diagrams", but with the latter, the more advanced topics in the Stanford article should also be included. Since we are already having more than enough difficulty with Euler and Venn Diagrams, I suggest the former. Dagme (talk) 18:40, 18 November 2009 (UTC)

I've cc'd the later of the above discussion over to Talk:Venn diagram with my responses there. Bill Wvbailey (talk) 20:29, 18 November 2009 (UTC)

First Order Logic
The following is from the introductory section:

"Set theory, formalized using first-order logic, is the most common foundational system for mathematics."

This statement is misleading at best, since set theory cannot be formalized within First Order Logic.

Dagme (talk) 20:32, 17 November 2009 (UTC)

ps How does one edit that introductory section?

Dagme (talk) 20:37, 17 November 2009 (UTC)


 * Wow, now here you actually have a point. That line has bugged me too &mdash; for one thing, it's not clear that mathematical foundations are ordinarily formalized at all, much less in FOL.  I'm not sure exactly what to do about it, though, since it does make an important point.  I'd like to say what that point is, but I'm not managing to come up with a good wording, which is why I don't have a proposal for a fix. --Trovatore (talk) 20:39, 17 November 2009 (UTC)


 * The sort of people who are interested in Foundational Studies (you know the kind I mean) are particularly interested in formalized set theory. To the extent that these people represent "mathematical foundations", those foundations are usually formalized. If you just mean chapter 0 of undergrad textbooks, then of course things are not formalized, but I wouldn't call that foundations of mathematics. &mdash; Carl (CBM · talk) 21:00, 17 November 2009 (UTC)


 * (1) To edit the first section, use the "edit" link at the top of the page. (2) ZFC is a first-order theory, and so it is perfectly possible to formalize set theory in first-order logic. &mdash; Carl (CBM · talk) 20:55, 17 November 2009 (UTC)


 * I rephrased the sentence to emphasize ZFC, making the reference to first-order logic implict rather than explicit. &mdash; Carl (CBM · talk) 21:01, 17 November 2009 (UTC)

General Problems
From the introductory section:

"The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known."

In fact, Cantor's set theory was informal and plagued by paradoxes, in response to which set theory was axiomatized.

My sense is that this article and those on Euler and Venn diagrams are systemically flawed. I don't know how prevalent this is in Wikipedia articles in the exact sciences, but I would think that a healthy approach would be for the consortium working on a topic to consult with experts on a regular basis.

The article "Diagrams" in the Stanford Encyclopedia of Philosophy, with which I spent some time yesterday, strikes me as a model for what a Wikipedia article should be: It is exhaustive, accurate, well researched, and fully documented. Can't we bring Wikipedia articles on these topics up to that standard?

Please note, I have generally been satisfied with Wikipedia articles in math and science. But the problems with the articles at hand are very serious.

Dagme (talk) 21:48, 17 November 2009 (UTC)


 * Whether Cantor's set theory was "plagued by paradoxes" depends on what you consider "Cantor's set theory" to be. This has been the subject of much scholarly debate.  Wang Hao is particularly known for the view that Cantor's viewpoint included a notion of limitation of size, which prevents the antinomies from attaching, even without any formal axiomatization.
 * The "picture" of sets has since become much clearer; there is no question of the antinomies in informal set theory motivated by the von Neumann hierarchy. My view is that the antinomies stem from conflating the intensional and extensional notions; Cantor in practice did not do this, as his motivations for considering sets did not come from logic at all, but from real analysis.  He also clearly had a picture of the ordinal numbers as sui generis objects that were not coded as sets.  Taking those two things together, you can easily suppose that he was actually very close to something like the von Neumann hierarchy. --Trovatore (talk) 21:59, 17 November 2009 (UTC)

Dagme: Could you make a list of the systemic flaws you see in this article? Frankly, I have little interest in the article on Venn diagrams, but I will work to fix problems with this one when they are presented.

As Trovatore points out, there is no consensus among experts about whether Cantor's later work is an example of inconsistent naive set theory, or an example of a consistent set theory that foreshadows the limitation of size method in ZFC. However, all that our article says is "The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s." which is true even if Cantor's work is inconsistent, a point on which this article is silent. &mdash; Carl (CBM · talk) 22:22, 17 November 2009 (UTC)

Dagme, It's simply the nature of the beast that Wikipedia's articles are wide ranging in quality. We strive for the quality of works like Stanford Encyclopedia of Philosophy, with varying success. But Wikipedia is a work in progress, we're doing our best, and you're welcome to help. As for consulting with "experts on a regular basis" not sure what you have in mind. Some Wikipedia editors are experts (depending on your definition of expert and to varying degrees of course). And experts are of course consulted by reading the published literature, or conversations with colleagues and friends. But if you mean something more formal than this, no mechanism for such currently exists. As for the specific sentence you cite, it seems fine to me. How would you suggest that sentence be reworded? Paul August &#9742; 22:32, 17 November 2009 (UTC)

"Could you make a list of the systemic flaws you see in this article?"

"Systemic" means essentially that the answer is "no". I will, however, offer some general recommendations --- see below.

"Frankly, I have little interest in the article on Venn diagrams, but I will work to fix problems with this one when they are presented."

I think that the way to correct the errors on Venn Diagrams --- an error identical to the one we have been discussing also occurs at the beginning of the section on Axiomatic set theory is to keep the statement in both sections brief and include a reference to the Wikipedia articles on Euler and Venn Diagrams for further information.

Unfortunately, the Wikipedia articles on Euler and Venn Diagrams are also systemically flawed. I don't think it is reasonable to take a hands off approach to those articles if you are going to be involved in correcting statements on the topic, wherever they may occur.

In any case, I think it is acceptable, even desirable, to mention that the diagrams in question are often (mistakenly) called Venn Diagrams, since that is a an actual fact. However, it is also essential to mention that these diagrams predate Venn, that a system based on them was published over 100 years earlier by Euler, and that the system that Venn produced is a further development and refinement of Euler's system which remedied some defects.

My general suggestions about these articles is first of all, identify who wrote them and what qualifications are possessed by those people. Then, try to work with those people and consult experts in the fields in question. For example, on diagrams, you could ask for advice from Frank Ruskey, and Jon Barwise. On Cantorian set theory, why not consult with Hallett, Dauben, and maybe Keith Devlin, who seems to enjoy interaction with the public and has written a well respected book on set theory?

Regarding the comparison with the Stanford Encyclopedia of Philosophy, I don't know much about the quality of their other articles, and I was just talking about this particular one, which I think is top notch. However, if Wikipedia adopts effective guidelines, I see no reason why its articles in general shouldn't match the quality of the Stanford Encyclopedia of Philosophy. How about giving author and consultant credits? This might induce prestigious names to contribute. The general quality already is good enough to cause experts to consider their name on a Wikipedia article a feather in their cap.

I have also been struck by a surprisingly contrarian attitude among wiki contributors. I think these authors and their products would get a lot more mileage if such attitudes were dropped.

Finally, regarding the articles on Venn and Euler diagrams, these articles have been substantially edited since I participated. However, they are presently hopelessly muddled. I do not get a sense that the authors really understand these diagrams. And some of the comments on the talk pages for those articles are hopelessly off base.

I don't have a solution for those articles, because all I can think is that they should be modelled on the Stanford article.

Incidentally: What makes the Stanford article so good? Here are some features I have noticed:

An apparent sincere desire to be truly informative. Omission of unnecessary peripheral comments, following the maxim "Do no harm". A thorough knowledge and understanding of the topics and a sure handed ability to make a coherent presentation.

I hope these thoughts help!

Dagme (talk) 05:30, 18 November 2009 (UTC)


 * OK, so to summarize: You think the article is bad, but the only two points you've made are (i) one that no one really cares much about, the naming of the diagrams, except that most of the rest of us want to use the common names and you don't, and (ii) another one where your complaint seems to be the claim that Cantor started set theory, which he in fact did.  Anything else? --Trovatore (talk) 09:47, 18 November 2009 (UTC)


 * I looked through the article fresh this morning, and the main issue I see is that it is short. In particular, many of the later one-paragraph sections would benefit from being expanded to two-paragraph sections, which would give a better sense of the focus and spirit of the field. Also the "some ontology" section could be expanded to discuss, in addition to pure sets, both urelements and proper classes. &mdash; Carl (CBM · talk) 13:01, 18 November 2009 (UTC)

Questions about definitions
The article on set theory is interesting for me because of the many lines of development that exist, no hint of which is given in the basic presentations in the Statistics textbooks I've seen. Nevertheless, I came to Wikipedia to understand better the basic definitions but was mostly unenlightened. Maybe the article could be enhanced with a little more fundamental detail in the early parts? Or maybe I just need some brief lateral coaching by the experts? 1. A set is said to be a 'collection of objects' or, in some versions elsewhere, 'distinct' or 'discernible' objects. What is an object? My dictionary has 9 different definitions, the first of which is "a tangible and visible thing". The other 8 don't seem relevant. So, could we have a set of intangible, invisible emotions or ideas for example? Concerning tangible objects, are any disqualified from membership, e.g. two red snooker balls that look identical for practical purposes? 2. The empty set, [0], is an important part of theory but, to many of us, it is nothing, exactly, and so cannot be considered for membership of anything, especially not a set of 'objects'. Doing so seems to contribute to Cantor's paradox mentioned by Staple (Discussion article: Self-contradictions in set theory), quoting: <>. Why must all sets have a complement set and why must the empty set be treated as an 'object'? 3. It seems to me that if a set is defined as it is often used in common parlance, as "items (or concepts) conforming to a definition", these difficulties are circumvented. Firstly, one can contrive a definition with which no thing or concept conforms, e.g. (facetiously) cubic balls for playing snooker. In that case, the empty set has an existence and identity related to how it was defined, and the set of all definitions could contain more than one empty set, each identified by its different definition. This seems much more philosophically satisfying than treating 'nothing' as a member 'object'. I await with interest to read the self-contradictions ensuing from this suggestion! Ajrc (talk) 16:49, 27 August 2010 (UTC)
 * You might like to ask the question on the Reference desk/Mathematics which is for questions about mathematics. This page is really meant for discussing how to improve the article.
 * You could construct a set theory which did not allow the empty set, but it would not be very useful as you could not then construct the intersection of disjoint sets. Further a set theory which did have the empty set but did not allow recursion with sets containing other sets would be rather limited as you could not have the Peano axioms to define the natural numbers. --Salix (talk): 19:56, 27 August 2010 (UTC)
 * Thanks and regards Salix. 1) Readers can improve articles, not just experts!  It's the initial definition I have difficulty with.  2) The care and intricacy of reasoning in Peano's axioms for natural numbers that your refer to contrast with the apparently casual use by mathematicians of the word "object" in the definition of a set.  I can think of collections of named items which may or may not be "sets" depending on your point of view on objects.  Seems like a big ambiguity to leave at the very base of mathematics and in this article but, admittedly, I'm out of my depth here.Ajrc (talk) 15:33, 13 October 2010 (UTC)
 * Set theorists usually consider only set theory without atoms. In that case the only"objects" that we allow as elements of sets are – sets. It may sound weird, but after all, if you start with no objects at all (because you don't know any set yet) you can still form the empty set. Once you have that, you can form the set which has the empty set as its only element. It has 1 element, so it's different from the empty set.
 * In practice one usually uses other "objects" as well. E.g. instead of defining the natural numbers as special sets, e.g. 0 = empty set, 1 = {0}, 2 = {0, 1} = {0, {0}}, 3 = {0, 1, 2} = {0, {0}, {0, {0}}}, etc., we can assume that the natural numbers are already given as "atoms", i.e. like the empty set they have no elements. Maybe these comments help you getting started. Hans Adler 16:38, 13 October 2010 (UTC)


 * Hi Ajrc For an alternative opinion as to how useful this viewpoint is to what you are trying to learn may be found in Pourciau, Bruce: The education of a pure mathematician. Amer. Math. Monthly 106 (1999), no. 8, 720–732. Let me know what you think of it. Tkuvho (talk) 17:26, 13 October 2010 (UTC)

Removed extraneous info from article lead
"Concepts of set theory are integrated throughout the mathematics curriculum in the United States. Elementary facts about sets and set membership are often taught in primary school, along with Venn diagrams, Euler diagrams, and elementary operations such as set union and intersection. Slightly more advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum."

This is not relevant. This is an article about set theory, and putting information about the US educational system in the lead of the article is the worst kind of US-centric bias. If you want a section on the teaching of set theory, showing the US approach along with that of other countries, go ahead. But US-specific information about the teaching of set theory does not belong in the lead of the article! Quantum Burrito (talk) 21:51, 23 November 2011 (UTC)
 * There's no reason to bring nationalities into the discussion. I think you're quite right that the material doesn't belong in the lead, but it wouldn't belong in the lead even if it were about education world-wide.  --Trovatore (talk) 22:09, 23 November 2011 (UTC)

Assessment comment
Substituted at 15:47, 1 May 2016 (UTC)