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

Question

   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"?

Answer 1

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.

Answer 2

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 :

• The existence of the empty set is an axiom of ZFC or not?

• Axiom of empty set

and more ...

Answer 3

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.