Everything and nothing

This is a response to Volkan’s comment.

Nothing

The mathematician says:

Theorem There is only one empty set.

Proof Two sets are equal if they have exactly the same elements. (This is a version of Leibniz’s principle of identity of indiscernibles, and has been absorbed into Zermelo–Fraenkel set theory as one of the axioms.) Now two empty sets E₁ and E₂ have exactly the same elements (viz., none at all), and so they are equal.

The psychologist (in this case, Brian Butterworth, an expert on development of mathematical concepts in childhood, in his book The Mathematical Brain) says:

Although the idea that we have no bananas is unlikely to be a new one, or one that is hard to grasp, the idea that no bananas, no sheep, no children, no prospects are really all the same, in that they have the same numerosity, is a very abstract one.

The poet (in this case, Eryk Salvaggio, from “Five Poems about Zero” (this extract taken from Strange Attractors, edited by Sarah Glaz and JoAnne Growney) says:

Who could have
no apples?

The apples must
leave traces
of their absence
in your memory—
bright green shining
in the palm of your hand
raised by a delicate elbow
to your mouth.

It seems hardly necessary to add anything. I will just say that, in my own experience, if I have to explain the empty set, I tend to do it in a way that resembles Salvaggio’s apples: for example, the set of even primes greater than 2. Though it is a logical impossibility, you could (I think) imagine that there were such primes; the proof that there are none is what shows the set to be empty.

The axioms of set theory allow us to talk about {x∈A:P(x)}, where A is a set and P a property; in other words, we can select all the elements of A which have property P and gather them into a set (there may be none of them). But we cannot have {x:P(x)}, the set of everything that has P. So it is natural to explain the empty set by constructing it as a subset of a set we already know, like the set of natural numbers in my example.

Everything

Bertrand Russell’s paradox, which he discovered in 1901, is easily stated:

Consider the set of all sets which are not members of themselves. Is it a member of itself?

Formally, if R = {x:x∉x}, then

• if R∈R, then it satisfies the criterion for membership in R, that is, R∉R;
• if R∉R, then it fails the criterion for membership in R, that is, R∈R.

Both possibilities lead to a contradiction. We are forced to admit that there cannot be a set having the properties of R.

This, together with another accepted principle of set theory, the Selection Principle, says that there cannot exist a set Ω consisting of everything. For if Ω were such a set, then we could rewrite the definition of R as R = {x∈Ω:x∉x}, and so it would also be a set.

In fact, this shows that there is no set consisting of all sets; but in the traditional foundation of mathematics on set theory, “everything is a set”.

Note the subtle difference between

Everything is a set,

and

There is a set consisting of everything.

Mathematicians have different responses to this situation. Some say, This shows that infinity is dangerous; we should eschew it altogether. Some (the vast majority) say, We now have a prescription for avoiding trouble, the Zermelo–Fraenkel axioms; we should stick to those. Some say, “Everything” is too useful a concept to abandon; let us say that “everything” is a class. Classes are just like sets, except that we are not allowed to include a class in another set or class. So it is OK to talk about the class R of all sets which are not members of themselves (indeed, the ZF axioms imply that this is the class of all sets, since no set can be a member of itself according to the Axiom of Foundation). The problem arises when we ask if R∈R, since R is not allowed to be in anything.

Category theorists sometimes claim that their approach to the foundations of mathematics avoids these foundational problems. They talk happily about “the category of groups” or “the category of sets”. My view (strictly as an outsider) is that it doesn’t. Both Mac Lane and Grothendieck addressed this problem, a clear indication that there is a problem; and a fairly recent interview with Pierre Cartier in the European Mathematical Society Newsletter supports my belief. He said,

Nowadays, one of the most interesting points in mathematics is that, although all categorical reasonings are formally contradictory, we use them and we never make a mistake. Grothendieck provided a partial foundation in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive. In this respect, he was pragmatic: categories are useful and they give results so we do not have to look at subtle set-theoretic questions if there is no need. Is today the moment to think about these problems? Maybe …

Conclusion

So the general mathematical view is that “nothing exists but everything doesn’t”; more precisely, there is an empty set, but there is no set of all sets. (The latter, if we wish to give it some kind of existence, is a “class”, a kind of second-rate status forbidding it from taking part in most mathematical activity.)

The existence of everything entails the existence of nothing; indeed, the existence of anything (any set A) entails the existence of the empty set (the set {x∈A:x≠x}). But not the other way round.

Right, I had better put on my anorak and go out now …