# Why do we have a problem about understanding the concept of the "empty set"?

The title seems quite bit more generalized, but I'm saying about the philosophers and mathematicians in the past who discussed about the concept of nothing, or the empty set. I'm currently studying set theory again, and I'm trying hard to understand concepts as total as possible. But I don't understand that why have we had a problem about understanding the concept of the empty set, and why there is an axiom called "The axiom of existence", which also called "The axiom of empty set".

First, let me start with introducing my understanding of nothing; Before define it, I have to introduce a sub-universe (I haven't found any rigorous way to explain it yet, but only informal way.).

Definition 1.   A sub-universe is a part of my(or our) perceptions.

The reason why I define this is to avoid some paradoxes(e.g. Russell's paradox) from The universe of discourse. In mathematics, we limit our perceptions or analysis by means of the axioms. The sub-universe has similar contexts, but we can pick smaller universe whatever we want if we adopt this concept. (be cautious that this concept can only be applied relevantly when the rules of inference or rules of thoughts satisfied. i.e., it must not contain any paradox.)

Now I introduce the definition of nothing;

Definition 2.   Let S denotes a sub-universe. We say a thing is nothing in S if and only if the thing dose not contains any element from it's sub-universe.

For example, assume that there is a box which contains a blue ball and a green ball. And then let us denote this as a set B, B={blue ball, green ball}. And define a sub-universe, S, in this situation which contains the set B. And now, I will remove 'the balls' from the box, which can be denoted by {}. Then the box is nothing in S.

From this view, if we can expand a sub-universe as total as possible, then I think that there is no problem with the concept of nothing. I certainly have no problem in understanding nothing, and so I really can't get any necessity on "The Axiom of existence". To sum up, my question is simply;

Question.   Why "The axiom of existence"?

• It's my sense that the people who have a "problem" with the empty set are numerous amateur philosophers on Internet discussion fora. By "amateur philosopher" I mean everyone with enough higher brain function to register a handle on a forum and bang on a keyboard. I include myself in that category. Actual professional philosophers understand that the empty set is like an empty bag of groceries. It doesn't have anything in it. Which means that it contains all the purple unicorns. Everyone understands this. Are there reputable philosophers who "don't understand the concept" of the empty set? Jul 24, 2014 at 20:04
• Related to my question earlier: philosophy.stackexchange.com/questions/9246/… I still find the concept troubling if you try to interpret it at all. But if you see it as just a calculus, which I suspect is what most mathematicians actually do, then you just "turn" off that part of the intellect that wants to see it as anything but. Jul 24, 2014 at 20:45
• If you are defining things in terms of your perceptions, you are not doing set theory. Jul 24, 2014 at 21:32
• @user4894 Of course professional philosophers understand the mathematically relevant properties of the empty set. However there are metaphysical properties of sets (and so of the empty set) that are puzzling: Having the members some set A actually has is sufficient for being A. It is highly controversial if concreta like tables have properties that are sufficient for their identity. But without doubt sets have such properties. Why is that so? This issue has been discussed by Graeme Forbes and others. Jul 24, 2014 at 21:45
• About the specific question regarding the null-set axiom (the "existential" axiom regarding the empty set) we need it into axiomatic set theory simply because we are working into a mathematical environment and not in a philosophical one. In a math theory of sets there are no concepts of "nothing", but only two "concepts" : set, i.e. every "object" in the "universe" of set theory is a set, and membership, i.e. there is only one "relevant" relationship between two objects of this universe : the membership relation. Nothing more. Jul 25, 2014 at 7:05

The empty set as a concept in ZFC is unproblematic as it satisfies formal properties.

It is problematic when we ask what this means in ontological terms; or as Parmenides pointed out we cannot conceive the void - the void is not; it doesn't trouble us that a name - the void - has been assigned to this allegedly unconceivable object. The name of the void, of course, is different to the void itself. We can percieve the name but not its referent the void.

We were to think of an empty set as a set that refers to the void, then it has no referent.

Its these two senses must be distinguished; only the first sense is a good description.

Some hints.

See in Wiki Empty set :

Axiomatic set theory : In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

There is already an axiom implying the existence of at least one set. Given such an axiom together with the axiom of separation, the existence of the empty set is easily proved. In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again, given the axiom of separation, the empty set is easily proved.

Philosophical issues : While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling [D.J.Darling (2004), The universal book of mathematics, John Wiley and Sons, p.106] explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."

You can see in MathSE, some post about the empty set :

and more ...

• But what reputable philosopher actually has trouble understanding the concept? Am I being too literal? I can understand the concept of unicorns just fine, I don't have to believe in them. Jul 24, 2014 at 20:20
• But why you think that you have to "believe" in it ? We do not "believe" in mathematics ... We define concepts, build theories, use mathematics to solve problem, to make computations (for bridges, for aircraft, for space shuttles, for atomic bombs, ..) but set theory is not theology. Outside math, the empty set has no use at all, like monads or substance-accidents outside philosophy. Jul 24, 2014 at 20:28
• That's exactly my point. OP claims that people "don't understand the concept" of the empty set. I claim that everyone understands it perfectly well. The only people who don't, are on Internet discussion boards. In other words I think the question answers itself. Even people who don't believe in the empty set understand the concept perfectly well. Who is it, exactly, that the OP is referring to when he talks about those who "don't understand the concept" of the empty set? Jul 24, 2014 at 20:33
• @user4894 Ok, I have to claim that I've also worried about my generalization, so mentioned in the first sentence of my post. But there is the axiom of existence which seems quite redundant to me as the reasons are in the post, so I thought it's because there were people who have some major problem with understanding the empty set concept. Jul 24, 2014 at 20:40
• I prepared a long answer and then read this and realized that there is no point in submitting my answer. Jul 26, 2014 at 5:05

A set is not a container but only its contentents. So one can take the position: Without contents no set. For modern set theorists there is no problem because they are trained to be attracted by obviously counterintuitive or counterfactual properties (like the axiom of choice in cases where a choice, i.e., a real distinction of one element from all others is impossible). But even the founders of set theory considered the empty set as problematic or not existent. In the following I quote some rarely known statements.

Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German the word set has the meaning of many or great quantity. Often we find in German texts the expression große (great or large) Menge, rarely the expression kleine (small) Menge. Therefore Bolzano apologizes for using this word in case of sets having only two elements: "Allow me to call also a collection containing only two parts a set." [J. Berg (ed.): B. Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag, Stuttgart (1975) p. 152]

Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non-empty set of less than two elements: "For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is element of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element shall be excluded completely for certain reasons, although it might be convenient for other investigations to fabricate such." [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig 1887, 2nd ed. (1893) p. 2]

Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [Bertrand Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]

Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P = O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146]

And even Ernst Zermelo who made the "Axiom II There is an (improper) set, the 'null-set' 0 which does not contain any element" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same Zermelo himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 March 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]

So it is all the more courageous that Zermelo based his number system completely on the empty set: { } = 0, {{ }} = 1, {{{ }}} = 2, and so on. He knew at least that there is only one empty set. But many ways to create the empty set could be devised, like the empty set of numbers, the empty set of bananas, the empty set of unicorns, the uncountably many empty sets of all real singletons, and the empty set of all these empty sets. Is it the emptiest set? Anyhow, "zero things" means "no things". So we can safely say (pun intended): Nothing is named the empty set.

• I agree with you that the set with no contents has some metaphysical issues. But doesn't that apply equally to any set? If I have an apple on my desk, that's an apple. Set theorists claim that there is also a set containing the apple, a set containing the set containing the apple, and so forth. Each of these are literally not really there. They're only abstractions. And for that matter, the set containing the empty set is not empty (it contains one element, namely the empty set) yet it is just as fake as the empty set. Moral of the story: Math isn't physics. Math is abstract. Feb 11, 2018 at 21:29
• @user4894 My opinion is this: A set is nothing but the being together of its elements. Therefore the singleton is an element and nothing else. Compare Cantor who did not distinguish between singleton and set: hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 43. In this picture the empty set is simply noting because the being together of nothing is nothing. The axiom of foundation (which is by no means essential for mathematics) changes this situation. It allows to errect an edifice of nothing built on nothing. Useless and without any reasonable application. Feb 12, 2018 at 16:06
• Quoting known crank Mückenheim is not going to help you understand set theory. Feb 12, 2018 at 18:57
• I had set theory beaten into me by some of the finest minds at our greatest universities. Are you trying to learn set theory from a crank? Or are you Mückenheim? Or just coming here to grind an axe of some sort? Feb 12, 2018 at 23:04
• ps -- If you are Wilhelm Mückenheim, surely you are aware that your ideas are well outside the mainstream. You could be right and everyone else wrong, but what are the odds? And how does resurrecting a five year old thread help you make your case? Feb 12, 2018 at 23:10

There are two separate questions here: why do we have difficulty with the empty set; and why do we need the existence axiom. The well-known principle of mathematical induction includes two clauses: (0) verifying a proposition for the case n=0, and (1) showing that if it is true for n=k, it is also true for n=k+1. Without the "base" case n=0, you just can't get started. Similarly, to be able to build more and more complicated entities in set theory, one needs a starting point (which seems to postulate very little - namely existence of the empty set - but enough to build most entities mathematicians are interested in via the other axioms).

The reason we have difficulty with the empty set are similar to the reason why mathematicians historically had difficulty with the number 0; see e.g., this answer.

My personal view is that any extension to the counting numbers 1, 2, 3… requires additional axioms, none of which is obvious. Von Neumann once remarked that in mathematics one does not understand things, one just gets used to them. That is certainly true of actual infinities, and the numerous additional rules to avoid their paradoxical consequences. I’m with Wittgenstein on this (a finitist). The first bite of the apple is the empty set. As soon as zero was admitted as a number, a new rule was required to prohibit division by that ‘number’. As far as I’m concerned, the empiricist’s motto is “Show me one!”

• thanks for the answer, dick atkinson
– user67675
Nov 22, 2023 at 10:32