Talk:Empty set

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 Field: Foundations, logic, and set theory

Does the empty set contain the empty set?[edit]

If not, we should be more careful with phrasing like

"For any set A: The empty set is a subset of A:"

and so on, because if A is the empty set, the statement doesn't hold. — Preceding unsigned comment added by 148.137.252.140 (talk) 20:48, 8 March 2012 (UTC)

The empty set does not contain the empty set as an element. The empty set is a subset of any set. --Daniel5Ko (talk) 22:10, 8 March 2012 (UTC)
The empty set is a subset of itself, but is it a proper subset of itself? Vincent (talk) 11:39, 10 June 2012 (UTC)
Write down a definition of "proper subset", then read it. You'll see that the empty set is not a proper subset of the empty set. Why do you ask? The article doesn't talk about proper subsets. --Daniel5Ko (talk) 23:31, 11 July 2012 (UTC)

Concatenation[edit]

What is the empty set concatenated with some set S, |S| != 0?

Well, what is concatenation of sets? —Tamfang (talk) 14:49, 10 June 2012 (UTC)

Oct 2002[edit]

I removed this entire passage, based on material first submitted by a non-registered Wikipedian:

The empty set is very simple; ironically so simple that many mathematics students (and even professional mathematicians!) have a difficult time applying it correctly.
For example, take the first feature listed above, that the empty set is a subset of any set A. If you look up subset, then you'll see that this claim means that for every element x of {}, re are some of my reasons:
  1. "The empty set is very simple" is not NPOV, simplicity being a far from simple concept. I've tried to compensate for deleting this by giving the briefest mention of how intuition can conflict with the formal definition of a set. (This discussion probably belongs in a different article, but...)
  2. The second paragraph seems, in context of the first, to be an attempt to describe a common difficulty in "applying" the empty set concept "correctly". However, it is more a justification for why the first property of the empty set listed by the article is reasonable. As such a justification, I think the second paragraph is underdeveloped, and worse than no paragraph at all.
  3. The second paragraph has an implicit discussion of vacuous truth and the third has an explicit reference to it. At present, I believe the vacuous truth article does a better job explaining these issues.

-- Ryguasu (Wed Sep 25 01:01:06 EST 2002)

The nonregistered Wikipedian is me.

  1. No argument; what you've written is good, better than what I wrote.
  2. I don't see why you don't want to have an example of what we're talking about. First, how is this example "undeveloped"? It goes on a lot more than any straightforward application of the definitions; to just prove the statement is trivial! This is a standard example of nonintuitive reasoning that I've seen in several set theory books, and it seems odd to consider it out of place. If anything, we should have more of this sort of thing in here.
  3. What you have doesn't even explain how the empty set is related to vacuous truth. That's a major omission! Not every fact must appear on only one page; most should appear on several, since they deal with several things.

Toby 08:54 Sep 27, 2002 (UTC)


I'm baffled by the bit about everybody, even mathematicians finding it baffling. It is terribly simple. The hard stuff is problems like deciding how many elements a set like { { {}, { {} } }, { } } has, and even that's just a case of keeping a clear head and counting the brackets -- Tarquin

I wrote "and even professional mathematicians!" back in the day because I see such confusion so often. Whenever I read math texts, I'm always on the look out for incorrect handling of degenerate cases in definitions and theorems. (Of course, one can argue about the definitions, but not the theorems!) The empty set pops up here all the time.

The most common example in my personal experience is this statement of the axiom of choice:

Every nonempty collection of nonempty sets has a choice function.

This has one "nonempty" too many (look at our article to see which, because we get it right). Removing the extra "nonempty" results in an equivalent statement, of course, since the empty collection has a choice function (the empty function). But it's presence implies that the word is necessary, giving a false impression; and it wouldn't be there if somebody hadn't had a false impression before.

If you give me a chance to look in some books, I'll find you examples where confusion about the empty set leads not just to misleading inclusion of unncessary words but to actual false statements. Finding and correcting these is almost a hobby of mine.

Toby 11:25 Sep 28, 2002 (UTC)

Examples of such mistakes would be an interesting addition to the article! (I had a lecturer at university who didn't believe in the axiom of choice... ) -- Tarquin

Hum, now that means that I have to find a selection of standard texts (it's no far just picking things randomly off the shelves) and searching for this particular error. That could take up a lot of time, since I'd naturally get drawn into reading the other parts of these books ^_^. (And I don't believe the axiom of choice either, at least not on weekends.) — Toby 08:18 Sep 29, 2002 (UTC)

Sorry, didn't mean to send you off on a research quest! -- Tarquin

Don't worry, I probably won't do it ^_^. — Toby 10:52 Sep 29, 2002 (UTC)

Question: does the empty set have a well-defined notation? For example, the set S = {1, 2, 3, 4, 5} = {x : 1 <= x <= 5} is well-defined. Does the empty-set have an analagous notation, e.g. ES = {} = {x : x \notin S} for any nonempty set S? The first part of this definition says that the empty set *does* contain elements, otherwise the set is not well-defined. The second part of the definition tells us how to construct the empty set; define a set, then remove all elements until the set is empty. But then we get a contradiction from defining the empty set as a subset of *any* set, because any of these subsets is defined as: {x: x\notinS, x\inS}. So is there a real set-theoretic definition and notation for the empty set or do we just conceptualize it for convenience? — Elijah Gregory


Regarding the difficulty/simplicity of the concept: the other day I talked to a math Ph.D. who said that he thinks of the empty subset of R2 as being different from the empty subset of R. AxelBoldt 23:54 Sep 29, 2002 (UTC)

Does he think of the empty subset of Q as different from the empty subset of R? How about the subset {0}? Or the subset Q?

I often think of the same set as different subsets, depending on what set I'm thinking of it as being a subset of, much as I think of the same set of ordered pairs as different functions, depending on what set I'm think of as the codomain. (And this is more than just an analogy, since subsets of X can be identified with equivalence classes of injections into X, and indeed it's that concept that is used to define subobjects in category theory.) This is a matter of keeping track of context, and is useful when answering questions such as "What is the complement of {}?". You can formalise it as an ordered pair (A,X), where A is a subset of X (just as you can formalise a function as an ordered pair (G,Y), where G is the graph and Y is the codomain). Then the question "What is the complement of A?", normally ambiguous, is unambiguous, because "A" really means (A,X) and you're just abusing notation when you call it A.

But if your colleague treats {} specially in this regard, then I'm afraid that I can't help him. — Toby 07:56 Sep 30, 2002 (UTC)

The question "what is the complement of X" is meaningless. It's an acceptable shorthand when we all know what the superset is, but "complement" is a 3-way relation (or a binary operation): "A is the complement of X in Y". I wonder, does Axel's friend think of the zero in R as different from the zero in the Q? -- Tarquin 12:25 Sep 30, 2002 (UTC)

First, how can you say that "What is the complement of A?" is meaningless if you agree that it's acceptable shorthand in certain contexts? In those contexts, it has a perfectly good meaning! That is why you'll often see sentences just like it in reasonable math books. All meaning depends on context, even in math.

Second, the 0 element of R is different from the 0 element of Q in much the same sense as their empty subsets are different. Even more so, because people that stick to pure set-theoretic reductionism in their math will still see them as different (unless they go out of their way to avoid this), which you can't say in the subset case. (Say, one 0 is an ordered pair of sets of rational numbers, while another 0 is a set of ordered pairs of integers.)

My point is that when you ask if two things in math are equal or distinct, the question only has meaning when it is relevant, that is in a context where the two things are already of a certain type (such as subsets of R2 or elements of Q). Yes, the question "Are they equal?" has a meaning in a reductionist set-theoretic sense (using the axiom of extension), but since the answer to the question varies depending on how one carries out the reduction, it is ultimately meaningless. Thus, it's best to regard such things as so different that they are "not even distinct" (as Wolfgang Pauli might have put it). Or to put it another way, you're asking if they're the same species, when they're not even the same genus. — Toby 10:23 Oct 7, 2002 (UTC)

"meaningless" in that it's not clearly defined.
But it is perfectly well defined if you specify ahead of time that you're dealing with subsets of X. If you like algebra, then you might want to ask yourself what is the cokernel of the group homomorphism sgn defined on the symmetric group S3, where sgn σ is 1 or −1 depending on whether σ is odd or even. Well, the answer to my question is not clearly defined.
For most purposes in algebra, however, you can work in a context where you assume that signs belong to the group {1,−1}, since these are the only values that signs can ever take in any permutation situation. Then the cokernel is trivial, since the image of f is the entire group {1,−1}. On the other hand, you could work in a context where you take signs to belong to the group U(1) of unit complex numbers (which is just what you want in some applications to quantum physics), and then the cokernel is an infinite group (in fact isomorphic to U(1) again).
So the answer depends on the context. Absolutely, the problem is ill defined if the context is not specified (or not specified enough). But by the same token, the problem is quite well defined if the context is sufficiently specified.
Indeed, it's become fashionable (thanks to Bourbaki) to define group homomorphism in such a way that the necessary context for the above problem must be specified. Wikipedia's own definition requires a homomorphism to be a function (obvious enough) and then defines functions to be differnt if they have the same domain and graph but different codomains. And the codomain is exactly the context that we needed here; it was (in my examples) either {1,−1} or U(1).
We don't do this with the term subset; if you say "A is a subset", then you don't need to specify what it's a subset of. A subset of Q and a subset of R2 could well be equal under this definition. Yet I hope that you realise how unsatisfactory it is to say simply "A is a subset". You want to say "A is a subset of X", or else you're not really finished with what you wanted to say. So we could (and doubtless would, had Bourbaki seen fit to so influence us) define a subset to be an ordered pair (A,X) such that ∀xA, xX, much as we define a function be an ordered pair (gr,Y), where gr is a set (of ordered pairs) satisfying a certain condition involving Y. Back in the day, they just said that gr was the function, but ultimately that's missing something relevant. Similarly, to say that A is a subset is not really enough to do any good; to get anywhere, you have to say what A is a subset of.
"one 0 is an ordered pair of sets of rational numbers, while another 0 is a set of ordered pairs of integers" -- yes, but once R is built, do we still consider Q to be the old construction of pairs of integers, or do we consider Q to now be a subset of R? I suppose there is an "original" Q and a "meta"-Q. Ouch! -- Tarquin
Yes, and you'll sometimes even see that written up in textbooks. The alternative method, which you will also see written up in textbooks, is to take the newly constructed set R, remove all of the metarationals from it, and replace these by the original rationals, thus getting a proR ("προ" being the opposite of "μετα").
The important thing to realise, however, is that all that you actually need to do mathematics is a field Q with certain properties (prime, charcteristic 0), a field R with certain properties (ordered, Dedekind complete), and a monomorphism (that is, both a homomorphism and an injection) from Q to R (there is only one). Exactly what the sets are is irrelevant, and it's even unnecessary to assume that Q is literally a subset of R; you can consider it as just an abuse of notation (we could use an article on that) when you pretend that it is.
This all makes perfect sense from the viewpoint of category theory, where the notion of subset doesn't generalise precisely to arbitrary categories but there is a notion of subobject — according to which, any monomorphism is in exactly the same position as an actual subset inclusion. That is, when you generalise to arbitrary categories, you can no longer tell whether you really have a subset or just a monomorphism, and it doesn't matter. — Toby 10:46 Oct 10, 2002 (UTC)
I suppose that is similar to the question: "is there only one group C3 or are there many groups isomorphic to it?" -- Tarquin

Right, although there is a subtle difference between the case of C3 and the cases of R and Q.

What is really odd is that there is a sense in which it's OK to speak of "the" cyclic group with 2 elements but not OK to speak of "the" cyclic group with 3 elements. The reason is that not only are any two cyclic groups G and H each with 2 elements isomorphic, there is also only one isomorphism between them. Thus, if you and I (discussing "the" such group with one another) come up with different groups in our respective imaginations, we can not only rest assured that there is an ismorphism between our groups but also that we know what that isomorphism is. For some purposes, we need that level of certainty to communicate with one another, a level of certainty that just isn't available with cyclic groups each with 3 elements.

To see this in a more familiar place, consider "the" Dedekind complete ordered field. It's reasonable to use this as the definition of the set R of real numbers, because if you have one set R (say a set of Dedekind cuts) and I have another set R' (say a set of equivalence classes of Cauchy sequences), then they are isomorphic. But more than that, they are uniquely isomorphic. Thus I not only know that when I say "√2" to you that there is something in your R that corresponds to my √2 ∈ R', but I also know that there's only one thing that it could be.

In contrast, it's less safe to define the set C of complex numbers as "the" algebraic completion of R. The reason is that there is no way to distinguish i from −i; they have precisely identical algebraic (and topological) properties. The result is that we have to set conventions, for every representation of the complex numbers, for which square root of −1 is the official i. For example, in the Argand plane, i is counterclockwise from 1, while −i is clockwise. But it could just as easily have been the other way around. In Schroedinger's equation, momentum is −ihd/dx, not ihd/dx; but it could just as easily have been otherwise. Every time complex numbers appear, this convention must be set, or conflicting implementations will result. No such problems arise with the real numbers.

Toby 10:06 Oct 18, 2002 (UTC)


I removed:

In fact, in set theory, all objects which don't contain themselves can be constructed from the empty set and the operation of making a set containing some available elements. The empty set acts as a starting point.

First, no set can contain itself, a consequence of the axiom of foundation. The claim that all sets can somehow be constructed from the empty set should probably be formalized as "all sets are constructible", a statement which can neither be proved nor disproved from the ZFC axioms. AxelBoldt 13:09, 29 Nov 2003 (UTC)

The empty set symbol[edit]

The letters Φ (obtained by typing &Phi;), φ (&phi;) or {}{}\phi (<math>\phi<<\math>) are not the symbol for the empty set in mathematics, and should not be used as such in Wikipedia.

In Unicode, the empty set symbol ∅ (&#8709;) occupies code point U+2205. But many fonts in use today don't include this character and render it as a small rectangle.

The TeX symbol \emptyset (<math>\emptyset</math>) looks funny and seems to dance above the baseline.

Therefore, I recommend using either Ø (&Oslash;) or {} ({}) to indicate the empty set.

Herbee 2004-02-18

I just found \varnothing (TeX \varnothing) which now looks best to me.
Herbee 23:18, 2004 Mar 18 (UTC)

I agree varnothing looks best. If we really want to avoid tex-png than Oslash seems the best choice. It also looks the most similar to varnothing. MarSch 13:46, 20 Apr 2005 (UTC)
The article Ø talks about ∅ (&#8709;). I think that this article should as well—or at least present the same symbol as a graphic. Perhaps there should be a new section at the end about representing the empty set in Unicode. BlankVerse 05:32, 3 May 2005 (UTC)
Mention &#8709; by all means. However, to be frank, I rather dislike all the sections in various articles about how to represent all kind of symbols in Unicode. I feel that this is a technical issue that has little to do with the concept empty set and is of little use to our readers. -- Jitse Niesen 10:01, 3 May 2005 (UTC)

"Boundings of the empty set"[edit]

I've moved the following newly added section titled "Boundings of the empty set" here for discussion

(Start of moved text.)

The empty set, like all sets, has a greatest lower bound and a least upper bound. The empty set and the real set differ in the bounds. The real set has a \sup\mathbb{R}=\infty and \inf\mathbb{R}=-\infty. The empty set is different though. In the empty set \sup\emptyset=-\infty and \inf\emptyset=\infty. This is logical because the upper bound is what is greater than or equal to any element in the set. For example, take x=5. Is x greater than or equal to all the elements in the empty set? Yes! Now, take x=-958. Is x greater than or equal to all the elements in the empty set? Yes! If the idea is extended to the entire real set then it is seen that the empty set has \sup\emptyset=-\infty. If the idea is applied to the greatest lower bound this time going from negatuve infinity to positive infinity then we can arrive at the same conclusion. The bounds property for the empty set is not an axiom, it is a theory.

(End of moved text.)

The idea that the empty set has (in some contexts) an inf and sup is an interesting property, which I think deserves including in the article. However, there are several problems with the text as it now stands. To begin with the first sentence as written is wrong, not every set has a greatest lower bound and a least upper bound. It is of course true that every bounded subset of R has a inf and a sup. And every subset including the empty set, of extended reals has infs and sups. This can be easily fixed of course. There are other quibbles I have with this section, which I can discuss if anyone cares to. But if not, I will just supply a rewrite, unless someone beats me to it ;-) Paul August 17:33, 14 September 2005 (UTC)

As a related question, isn't it that inf and sup are defined for non-empty set of real numbers? We can talk about Z (set of integers) because it is the subset of real numbers, but the empty set is a subset of any set, not necessarily R. Our Supremum is not clear on this issue, though it should be. Anyway, this is interesting; I've never thought of this case. -- Taku 23:04, 14 September 2005 (UTC)
== Boundings of the empty set ==
The empty set has a greatest lower bound and a least upper bound. The empty set and the real set differ in the bounds. The real set has a \sup\mathbb{R}=\infty and \inf\mathbb{R}=-\infty. The empty set is different though. In the empty set \sup\varnothing=-\infty and \inf\varnothing=\infty.
For direct proof, suppose that x\in\varnothing and suppose that y\in\mathbb{R}. By definition of the upper bound it can be seen that (x\le y)\forall(y\in\mathbb{R}). Thus \sup\varnothing=-\infty.
The proof for \inf\varnothing=\infty can be followed by the same logic. The bounds property for the empty set is not an axiom, it is a theory which can be proved by the axioms of the bacis real number set.
I'm sorry about all that. I'm just an undergraduate at University of South Florida. My professor told me about the property of bounds on the empty set on Monday (September 12, 2005) so I decided to put it up. I don't really know anything about math or all that but I saw it wasn't in the article so I wanted to put it up. I can get a proof of the entire thing (if the one I redid above isn't right or is too short or something) from my professor. He'd be really happy to help I think. Sorry about it all again. ~Econ Schol
I am not questioning your proof. What I am wondering is about definitions. To me, talking about bounds of the empty set sounds strange. Also, the consequence of this, that is, sup may be less than inf may not be desirable. On the other hands, defining sup and inf for the empty set as well might prove to be useful in proof-writing (same idea as zero factorial). -- Taku 00:59, 18 September 2005 (UTC)

I've seen this used in measure theory. Your proof is flawed though, since "suppose x is in the empty set" is a contradiction and thus allows any conclusion. --MarSch 09:47, 25 October 2005 (UTC)

Bourbaki and Ø[edit]

It seems clear that the Ø symbol was introduced in Éléments de mathématique (Bourbaki, 1939). The question is how much background info to include in this article on Bourbaki as opposed to Ø. Of course, we should include the specific information (which was, by the way, previously missing) that the symbol was introduced in 1939. But what is the point of mentioning that Bourbaki was a group mostly French mathematicians of the 20th century, when this information is a click away and is common to everything else mentioning Bourbaki? That's the beauty of wikilinking. Otherwise, why not include the Axiom of empty set here? or even the definition of a subset? --Macrakis 00:00, 25 October 2005 (UTC)

Whoa there![edit]

I just found this page, and I've removed the text:

  • The empty set is NOT 0.
    \varnothing ≠ 0

To the contrary, it is often useful to define 0 to be the empty set. Why was this italicized, bolded, in all caps, and at the top of the Properties section?! Melchoir 01:13, 15 December 2005 (UTC)

Upon browsing ths history, it looks like 80.179.80.156 did it. Sigh... Melchoir 01:15, 15 December 2005 (UTC)

The number zero is the cardinal number of the empty set. That is the number of objects in it. The empty set itself is not "nothing," it is a set. Gregbard 10:05, 2 August 2007 (UTC)

Iit_bpd1962 comments: 29 April 2006[edit]

I differ in the semantics of: "The empty set is not the same thing as nothing"; it is a set with nothing inside it, and such a set exists, which is the starting statement of this article. That is has suppoting ZF logic axioms is not comforting since on questioning axioms of Logic as extant today is itself being questioned and if not then is must be questioned.

If one can show that the notion of nothing is indeed required in the argument which claims "nothin can exist", then "nothing exists" must be entertained as a TRUE statement. However, "noting exists" is itself self contradictory. It says, that there is something like the notion of nothing and that this notion of nothing exists.

Now imagine "nothing exists" as TRUE. Then it is claiming that In the universe there is a consciousness (and this is neccessary, otherwise "who", "what" is making this / observing to clain "nothing has benen cognized to exist?) Thus even if allow "nothing exists" to be a vlid allowable notion we a forced to also say that a cognizer is also present. But then "nothing exists" itself is false.

I thus must beleive, that such statements of THE null set not being nothing are in this sense FALSE. The null set is indeed the same thing as nothing, arrived at because consiousness as a pre-requisite for stateting any matters about existence was not takn into account. If it is further argued that there is a cognizer and the cogniser is the one stating that the null set exists and that the null set is indeed not nothing, then it follows that the null set has no elements. Then what is this cognizer entity, cognising? It is cognizing the null set having no elements. This is physically meaningless and is thus impossible. So, what is platonic paradigm being formed which itself uses reality, which may not itself subscribe completely to platonic notions. (see discrete and continous mathematics).

Such claims are arrived at through sheer inacceptence of a material world which is finite (and discrete) and required to advertise such claims in the fist place. So what is being indulged in is that simultaneously two notion are made which are contradicting each other. Lesson: all atomic axioms of a Logic system must acknowledge consciousness as a required cognising / advertising entity, which itself cannot be made "zero", for if it made zero then nothing can even be uttered, leave alone "established".

{This topic is to be shortly continued, in which a new atomic axiom set may be formed overriding many ZF, Russel's (new) non-paradox axioms will be examined and proven not be really axiomatic and in tune with consiousness)

symbol[edit]

Not sure whether it's an exclusively local problem, but the whole section about symbols doesn't render properly here (it looks like my browser substitutes a zero-with-slash glyph for the empty set codepoint). Would it make sense to include an image of what precisely the empty set symbol looked like when Bourbaki used it, and maybe tone down the rest of the section a bit to clarify that many people, of course, use slight variations of the Bourbaki symbol?

RandomP 06:17, 4 May 2006 (UTC)

"the"[edit]

Mathematicians speak of "the empty set" rather than "an empty set". In set theory, two sets are equal if they have the same elements; therefore there can be only one set with no elements.

But mathematicians also speak of "the trivial group", "the cyclic group of order n", and so on. There's certainly no harm in using the definite article for objects in a category with a trivial automorphism group.

I think this does make the above paragraph a bit misleading, at least.

RandomP 06:25, 4 May 2006 (UTC)

Pun[edit]

The great thing about mathematics is that it allows one to be upset over nothing.

^--That's really a pretty good one.

Tom McKay et al.[edit]

I recently made a rather long few posts on the discussion page for the article on set theory that raise some of the same questions as those discussed in the section on whether the empty set exists or is necessary. An author, Tom McKay, is mentioned and a page is given for his university, but I am having trouble finding the book mentioned in this article (no title is given, for one thing). If anyone knows the title of this book, and, even better, how I might go about obtaining a copy (I didn't see him in a search on Amazon, for example), I would greatly appreciate this. It might also be nice if someone included the title in the article.

To me, the empty set is not simply counter-intuitive; it seems illogical. If a set is defined as a group of elements, then something with no elements is, as Lowe apparently puts it, a non-set. Why can't this be so? Wouldn't Occam's Razor suggest that we eliminate "empty sets" from a theory and eliminate the distinction between membership in a set and being a subset of it? That an empty set is an element of every set makes it also an element of the complement of every set, so it is both within and outside of every set (if it even exists). This is a contradiction in terms.

We have, as far as I can see, simply assumed some "concept of the set" without clearly defining it. A bag is not a useful analogy here because it is a real object that has real properties, and thus can be approximated in a model by a set of all of its observable properties. There is no such thing as something with no observable properties. See the article on the thing in itself and its criticisms: it seems to me that the empty set shares a lot with it.

Furthermore, in what I've read the very notions of the set and of membership in sets have been described as intuitive anyway, so how can we even discuss their properties precisely without examining their origins first? We have forgotten to quantify, in real terms, the difference between membership in a set and being a subset of it except in terms of sets themselves, which are in turn defined by the properties of their members and subsets, et cetera. It looks like a circular definition that leads to the proposition of a thing whose only property is that it is that thing (i.e., something utterly indeterminate and inobservable).

I'd like to read about the alternative suggested by McKay, simply to see whether it is at all viable. If anyone knows it, please post the title of his book. Tastyummy 04:10, 23 August 2006 (UTC)

Tastyummy: the empty set is a subset of every set, not an element of every set. There is a huge difference. You should take your questions to Math Stack Exchange. Also, it is hard to imagine working with sets without ever using the empty set. It would be like trying to do algebra without ever using the number zero. Gsspradlin (talk) 23:49, 31 August 2013 (UTC)

I don't know anything about McKay, but.... There are certainly many different ways to define mathematical concepts. The choice of standard definitions depends on many things: partly the accidents of history, partly aesthetic and stylistic considerations, but above all by what works well in mathematical practice. This is no place to give a whole summary of the history of set theory; suffice it to say that all standard set theories distinguish clearly between subsets and members, and include the/an empty set. If you have good evidence of a notable alternative (I don't know of one), discuss it here, but keep in mind that there are always fringe theories which are not notable in the WP sense.... For more on the history of mathematical concepts, you might want to read Imre Lakatos's great book Proofs and Refutations. Good luck in your mathematical explorations. --Macrakis 17:57, 24 August 2006 (UTC)

The reason I've asked for the title of this guy's book is that I'd like to see for myself whether it's "notable". I understand that it's considered a "fringe theory", but that doesn't mean I won't even glance at it. Of course I don't have any evidence for an alternative: but I'd like to at least consider what might be evidence, even if it's probably just some nobody making things up. I can't help but find the empty set illogical; I'll note here that I'm certainly not only investigating weird quasi-set theories, but that I am trying to learn about ZF set theory, etc., and that the only reason I'm looking into theories that claim not to make use of the empty set is that I can't yet see why it's necessary: I have to figure this out before I can move on. I still can't see what's wrong with Lowe's suggestion that we call things without elements "non-sets" instead of "empty sets", since a set is supposedly a group of elements. What would be lost in doing this? (I'm not saying nothing would-- I'm just saying that I don't yet know why the empty set is necessary.)
I am trying to learn mathematics from the bottom up, and if the definition of the empty set describes it both as a group of elements (a "set") and not a group of elements, it seems self-conradictory, and this is hindering my progress. I don't see how there can be a "group" of zero elements. In order to have a group, as I understand it, we must be able to categorise things according to their similarities and differences. What similarities are there between an utter lack of anything that can be conceived of as an "element" and anything else whatsoever? Again, why not just call some things sets and others (i.e., lacks of elements) non-sets? I don't mean within formal set theory as it stands; I mean in using mathematics as a model of reality. And if we are to consider all mathematical objects as sets, then why aren't elements to be considered as sets? This would make any element of something also a subset of it. Again, I know this isn't how any current set theories work: I'm just wondering why this must be so.
I'll certainly take a look at Proofs and Refutations, as you've suggested. Thanks for your attention and patience.

Tastyummy 00:53, 25 August 2006 (UTC)

Just because something "seems illogical" or "seems self-contradictory" to you or me is not terribly relevant; Wikipedia reports on what mathematics is, not what we think it should be. On the other hand, it is true that it would be valuable to explain the background and motivation of concepts, both historically and philosophically, but that is hard. Let's find some good sources (cf. Wikipedia:No original research).

You mention that you want to "learn mathematics from the bottom up". I'm afraid that is a bit of a chimera. The so-called foundations of mathematics claim to be logically prior to the rest of mathematics, but unlike the foundations of a building, they were actually created after much of the mathematics they are the foundations of. Better to start with mathematics 'in the middle' and work outwards to more abstraction.

Finally, a few quick substantive comments on your problems with the empty set. If you think it is illogical to talk of an empty set, how about the number zero? Numbers, after all, are supposed to count things. How can you count no things? Should zero be considered a non-number? Think about intersections of sets -- the set which represents what two other sets have in common. The intersection of {a,b} and {b,c} is {b}. What is the intersection of {a,b} and {c,d}? You could say it was undefined (like 1/0), but wouldn't it work better if you could treat the result as a full-fledged set, assuming you could do that in a consistent way (which you can)?

Good luck in your continued studies. Read Lakatos and you'll be ahead of most college students (and maybe even many professors of mathematics) in understanding how math really works -- what all those definitions and axioms and proofs do for you. --Macrakis 18:44, 25 August 2006 (UTC)

Thanks again for your attention. I wasn't suggesting that the articles on set theory be modified in any way to reflect that I personally am having problems with certain concepts in it-- I'm just trying to learn about those concepts; I agree that Wikipedia articles on mathematics should reflect accepted and current formulations, and I'm sorry if I gave the impression that this wasn't the case. The article is, as it stands, valuable and interesting.
As for the number zero: this, to me, is a much more tangible concept than the empty set. I have zero elephants in my apartment. But I disagree with the statement made by user:tezh when he was trying to explain the empty set to me that "the set of elephants in my apartment is empty": I disagree for the simple reason that, the way I see it, there simply is no set of elephants in my apartment. I'd say that two sets that share no elements simply don't intersect, rather than saying that their intersection "is" the empty set. Again, I don't see what's wrong with Lowe's statement that collections of elements are sets, while things that are not collections of elements are simply non-sets. I still can't see the problem with this. I do not understand how there can be a group of zero elements, even though I understand that the number zero is real and useful, because all that is left here is the "concept of being a group", which, to me, requires that there be members in the group. There would be no "concept of planets", for example, if there were no planets to speak of. For us to suggest that there must exist "a planet which is not made of matter and does not orbit any star, and which is not roughly spherical and whose path is extremely irregular" is clearly absurd: how is it not equally absurd to propose the existence of a collection of elements with no elements?
But again, I don't mean to imply that this article ought to be modified in any way. I'm just trying to learn for myself why this concept is really necessary. I will certainly begin reading the book you've suggested, and, again, thanks for your advice and for your time.
Tastyummy 19:55, 26 August 2006 (UTC)

It's not surprising that I thought that you wanted to modify the article, since the function of Wikipedia Talk pages is to discuss edits on the article. They are not the appropriate forum for background discussion of the concepts. You might try newsgroups like sci.math, though frankly I think you'll learn more from Lakatos. By the way, it is deceptively simple. Read it the first time without the footnotes -- it's a pretty easy read until the last chapter. Once you've understood that, then go back and read with the footnotes; amazing! --Macrakis 20:05, 26 August 2006 (UTC)

I'm not blaming you; I did definitely get off topic in the above discussion. But I did propose one change in my first post (that the actual title of McKay's book be added by someone who knows it, since the article currently only says that it's a "recent book" and it would be useful to actually know what book it is.)
Anyway, I apologise for wasting space; I just did the same thing on the talk page for set theory in general, and I realise it's not what they're for. I won't continue to do this.
Thanks again for your time and advice; I'll definitely look for a copy of Proofs and Refutations and begin it as soon as possible.
Tastyummy 22:04, 27 August 2006 (UTC)
I've just found something that may help another user in the predicament in which I found myself while writing the above confessions. From Boolos' Computability and Logic, first page of first chapter:
"By courtesy, we regard as enumerable the empty set, "\emptyset", which has no members. The empty set: there is only one. The terminology is a bit misleading: it suggests a comparison of empty sets with empty containers. But sets are more aptly compared with contents, and it should be considered that all empty containers have the same, null content."
Yeah, it's an introductory text, but it's Boolos, and it helped me (my private contention had been that in order to be a set, a thing was necessarily also a set of things (yes, specifically of a plurality)). It is also worth mentioning that while desired results in maths may be produced without the empty set (i.e., without using the words "empty set", "null set", the symbol "\emptyset", etc.), (because you can use any other symbol[ism] you like), they needn't be. If you call it "the non-set", or something similar, to yourself, you'll find you get the same results as you would otherwise have gotten, under interpretation. It helps, if you're a skeptic, (though I doubt there are many, if any, people by now as convinced of the "seeming illogicality" of it as I was), to study logic, and then use the axioms of set theor[y/ies] as they have been formalised symbolically as definitions (or assumptions, or hypotheticals, etc.) and see what happens. I hope this helps someone who took my earlier rantings on the empty set seriously, and again thank each user who recommended good reading and tried to coax me out of bashing a perfectly useful mathematical object's existence, necessity, or whatever.

-Tastyummy —Preceding unsigned comment added by 12.11.93.150 (talk) 07:41, 26 September 2007 (UTC)

Set theory history of { }[edit]

{ }: Null or empty set Ø: Is a major source of confusion. Zermelo called it "0":

"Axiom II. (Axiom of elementary sets) ... There exists a (fictitious) set, the null set, 0 that contains no element at all.... 6. The null set is a subset of every set M: 0 € M ..."(Zermelo 1908a:202).

"0" was also used by Frankel 1922 and Skolem 1922 to indicate the empty set { }. But this usage is risky because, for example, "0" is a valid symbol -- distingushable ink-marks on a page -- in the place adjacent to this word 0 , whereas the empty set { } means that, truly, the place between the brackets has no ink in it, no symbol-content whatever. (cf Hilbert 1904 where Hilbert started with | and adjoined it to || etc; Peano 1889 started from "1"; also Turing 1936-7 page 135 in Davis 1965 The Undecidable where Turing discusses the notion of "symbols on paper"). "O" used by von Neumann 1925. Nowadays "Ø" seems to be the preferred symbol (Halmos 1960, Suppes 1972, Enderton 2001, Boolos-Burgess-Jeffrey 2002) . Another example: In 7-bit + parity ASCII teletype code the NULL is 0008, i.e. NULL represents no activity on "the line" -- "the line" being a place in space (a wire) and "a symbol" is a certain type of activity "on the line -- whereas the code (the core serial code, not counting start, stop and parity bits) for symbol "0" is octal 0608 = 4810 = 0,110,0002. In the telegraphy-teletype world the line is always "high" (I think its actually -15 volts or so) meaning that the line is "okay" and not severed; the start bit sets things going, followed by the code 0110000 followed by a parity bit [positive or inverted] followed by one or two stop bits. Maybe someone knows if the null is transmitted this way:

null? .....111111111111110111111110111111111111111111.....
zero .....111111111111110011000000111111111111111111..... (i.e. "start=0", "0" = 0110000, parity=0, stop=0)

Or is "null" just the "high-but-inactive line" represented here by ...111111111111111111... ? This would truly be "null", i.e. no activity in a place called { }.

The papers of Peano 1889, Hilbert 1904, Zermelo 1908, Frankel 1922, Skolem 1922, and von Neumann 1925 are to be found in van Heijenoort 1967. wvbaileyWvbailey 14:50, 19 September 2007 (UTC)

Etymology[edit]

On the history of the terminology "empty set", see [1], "Earliest Known Uses of Some of the Words of Mathematics". Not related to the above discussion but, again, helpful for those of us who get confused by things like this.
From midway into the section on "set":
"Dauben describes the emergence of the Menge terminology on p. 170. In his early work Cantor used the term Mannichfaltigkeit, Riemann’s term often translated as MANIFOLD. The terms Menge and Mengenlehre appear in a note to the article “Über unendliche, lineare Punktmannichfaltigkeiten, (Part 5) Mathematische Annalen, 21 (1883), 545-591, issued separately as a pamphlet Grundlagen einer allgemeinen Mannichfaltigkeitslehre. The meaning of the term Mannichfaltigkeitslehre is discussed in a note on p. 587 of the article. Cantor distinguishes this general theory from geometrical Mengenlehre and goes on to explain, “By an ‘aggregate’ [Mannichfaltigkeit] or ‘set’ [Menge] I mean generally any multitude which can be thought of as a whole, i.e., any collection of definite elements which can be united by a law into a whole.” Dauben’s translation.

However Cantor did not adopt Menge and Mengenlehre as the terms until later. Both are used in his “Beiträge zur Begründung der Transfiniten Mengenlehre,” Mathematische Annalen, 46 (1895), 481-512. The opening words are:

   By a “set” [„Menge”] we mean any collection M of into a whole of definite, distinct objects m of our intuition or our thinking (which are called the “elements” of M) of our perception or of our thought."
Although it matters little, if at all, to most readings of the symbol {} or phrase "empty set" in mathematics, this is useful etymological information to people interested in it and I suggest that a link or excerpts from the aforementioned site may be appropriate in the article. If there are no objections, I'll put together a brief section on the etymology of the terms "empty set", "null set", "null class", etc., and hopefully without getting anything Wrong.
If there are objections to this proposal, now's the time to hear them. "Empty set" appears to have been used to mean several different things by several different mathematicians, not least among them in mathematical achievement Cantor, who, to reiterate, is quoted (and translated) thus: "“By an ‘aggregate’ [Mannichfaltigkeit] or ‘set’ [Menge] I mean generally any multitude which can be thought of as a whole, i.e., any collection of definite elements which can be united by a law into a whole.”"
Perhaps some of the confusion I experienced over the empty set could have been helped by a more thorough demonstration that its definitions have changed, and continue to change, over time. (Perhaps those demonstrations were made, albeit more tacitly, and I fail even now to notice them, but I don't see why the above text isn't notable, verifiable, etc. regardless of one's own most oft-employed axiomatisation, informal instructional text, etc., regarding set theory.)
Tastyummy; 26 Sept. 2007-- I am purposefully not signing because I haven't modified or checked my user or talk pages recently and will have them up to date, and a real signature here, soon.

---

The translation of Cantor (1897) that appears in Hawking 2005 And God Created the Integers, Running Press, Philadelphia, pp. 965ff shows Menge translated as "aggregate", and a "collection into a whole" as Zusammenfassung zu einem Ganzen. I tried to investigate Cantor's use of the notion "0", but it is not easy to find in this collection that has to do with transfinite numbers. I was under the (unfounded, unexplored) impression that he avoided 0 altogether, as did Peano in the true axioms (see van Heijenoort 1967:94) that Peano kind of cc'd from Dedekind -- they all started with | , the first of the natural numbers (signified by the symbol we call "one" and write as "1"). But, when I go back and look at Cantor's 1899 Letter to Dedekind (van Heijenoort 1967:113ff) I see that I put a little yellow tag next to the following:

"...when I speak simply of numbers, I shall have in mind only ordinal numbers, that is, types of well-ordered sets.
" I now consider the system of all numbers and denote it by Ω .... [he descirbes a proof here that ...]
" Ω is therefore a simply ordered system.
" But it also follows easily from the theorems on well-ordered sets proved in §13 that every multiplicity of numbers, that is, every part of Ω, contains a least number.
" Hence the system Ω, when naturally ordered according to magnitude, forms a sequence.
" If we then add 0 to this sequence as an element -- putting it first, of course -- we obtain a seqeuence Ω':
"0, 1, 3, 3, ..., ω0, ω0+1, ..., γ, ...,
"in which, as we can readily see, every number is the type of the sequence of all elements preceeding it (including 0). (The sequence Ω has this property only from ω0 + 1 on <<in fact, from ω0 on>>.)" (italics in original, paragraphing as in the original, << >> indicates a note added in the text, van H 1967:115)

wvbaileyWvbailey 21:10, 5 October 2007 (UTC)

Does it exist, or is it necessary? Section

The opposite of a great truth is also a great truth - T. Mann

For example, the opposite of empty set {} is a non-empty set, such as the set of integers, or finite sets we experience in our mundane lifes.

Also a lesser context can define the greater context, and vice versa.That is, each is defined by what it is not i.e. its antithesis; for example, the antithesis of quanta and spacetime manifold.

So what if a physical model such as our universe, or a divergent cyclical set of universes (hence non-empty set with 1:1 correspondence to integers), has a greater context of simplest case i.e. empty set? Might this be indirectly inferred; of course without perturbing such alleged greater context? Such as if there were multiple 'universes'.

So can {} be introduced into consideration of a description of a larger context?[2] Zanardm 04:27, 20 October 2007 (UTC)

Zero[edit]

There's a particular line from the article that I'd like to clarify, but I'm not sure whether my changes would fit with other views on the subject. It is:

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, zero is defined as the empty set.

The problem, as I see it, is that in most contexts it's not common to define the zero as the empty set (in fact, IIRC, even in the set theoretical definition of natural numbers, it's most common to define zero as the equivalence class of sets with the same cardinality as the empty set). As it is, it's hard to teach undergrads the difference between zero and the empty set, and it doesn't help if the wikipedia page on the subject is potentially misleading. I'd like to go with something like:

The connection between the empty set and zero goes further, however. Modern mathematics is based on set theory, and even seemingly intrinsic things, like the natural numbers, can be defined using sets. In this sense, the fact that \vert\emptyset\vert = 0 is actually the definition of zero.

If this wording isn't acceptable, I'd at very least like to somehow point out that in most contexts, \emptyset\neq 0. James pic (talk) 16:24, 5 August 2008 (UTC)

The Empty Set cartoon'd[edit]

See:

Strayan (talk) 03:22, 8 August 2008 (UTC)

Empty set is not an element(member) of any set.[edit]

I just thought it might be informative to add to the article that the empty set is not a member(or element) of any set. The empty set is only a subset of all sets.

This seems a small, yet important, thing to mention. 74.183.138.133 (talk) 01:31, 19 March 2009 (UTC)

I don't follow. Surely the empty set can be a member of another set. For example, the singleton set that just contains the empty set: {∅}. Ezrakilty (talk) 11:49, 19 March 2009 (UTC)

Yeah, I have to agree with the last person, the empty set can easily be an element in another set (often refered to as a family when it's a set of sets). As he pointed out, {{}} is the set containing the empty set. You could, of course, have things like {{}, {1}} as well. In fact, the empty set is always an element of the power set of a set, e.g., P({1,2})={{},{1},{2},{1,2}}.

Why Ø?[edit]

We all know about Ø, and that it was created by the Bournaki group, but I've never seen a good explanation of why it was chosen. Was it just chosen as special zero? The fact that it's the same as the Danish and Norwegian symbol is simply a coincidence, no? I mean, there's no word such as "Ømtig" that means empty, is there? —Preceding unsigned comment added by 97.124.76.43 (talk) 23:02, 27 October 2009 (UTC)

See [3] — Carl (CBM · talk) 19:48, 28 October 2009 (UTC)

{Ø} in a textbook[edit]

I have here a textbook, "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai. I think you'll like this quote:

"The empty or null set is by definition the set that contains no elements. This set will be denoted by {Ø}."

This is not a typo: the authors go on to use the same notation throughout. Does anyone else have an example of a book or other legitimate source that uses this notation? —Preceding unsigned comment added by Jalanpalmer (talkcontribs) 03:22, 4 November 2009 (UTC)

I don't know if it's a typo, but it's incorrect. The authors or editors made a mistake. {Ø} is the set containing the empty set (and no other elements). Ø is the empty set. They are not the same thing. I have never seen anyone else use this notation, in many years as a mathematician. Sometimes (not often) the empty set is written as {}. (this is correct). Gsspradlin (talk) 23:41, 31 August 2013 (UTC)

Proof of existence[edit]

Somebody I know defines a "set" in a way that requires it to have at least one element (thus, {a,b}∩{c,d} has no answer, akin to dividing by zero). Is it possible to produce an absurdity from this axiom and thus prove the existence of the empty set? --Shay Guy (talk) 20:10, 17 September 2010 (UTC)

Well, if the empty set is not a set then the distributive law no longer works for a start. If B and C are disjoint sets then
(A \cup B) \cap (A \cup C)=A
whereas B \cap C is undefined, so
A \cup (B \cap C) \ne (A \cup B) \cap (A \cup C)
Maybe you could define a consistent algebra of sets in which the empty set is not a set, but it would be very untidy, as every law would require exceptions and special cases to exclude the empty set. Gandalf61 (talk) 20:18, 17 September 2010 (UTC)

Section "Operations on the empty set"[edit]

I'm not so sure about this section:

Operations on the empty set
Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the identity element for addition, and one is the identity element for multiplication.

is "sum of elements" and "product of elements" trying to say nullary union and intersection? I don't think it's quite correct the way it stands. And the musings about 'says more about the operation in question' and 'are unusual' are probably non-neutral and could be profitably excluded. BrideOfKripkenstein (talk) 15:19, 4 May 2011 (UTC)

Discussion of visual representations of the empty set and nothingness[edit]

The empty set is the set containing nothing, no elements.
Alice: "I see nothing."
Cheshire Cat: "My. You have good eyes."[1]

Lewis Carrol was a logician who considered the concept of nothingness in the late 19th Century. "Blank" visual and aural images have been used to represent the concept of "nothingness' in the arts and applied aesthetics, as in Minimalism. Images An image or representation of "nothing" via the empty set was developed in the late 19th and early 20th Centuries, and remains a subject of controversy.
Does this image have more or less "nothingness" than the black image above. Aural and visual representations of nothingness occur in mathematics and in the arts. In the aesthtics of minimalism, a canvass may be left white or "blank", or it may be painted black. The blank white canvass reflects more light than the black canvass, but the black painted canvass has more paint. Which is a better closer visual representation of nothingness" is a matter of debate.

There is discussion of visual representations of the empty set and representing it at the Nothingness article. PPdd (talk) 15:52, 9 June 2011 (UTC)

  1. ^ Alice in Wonderland, Lewis Carrol