The Empty Set

With the axioms of extension and specification, we can begin to construct some basic set theory!

Existence of the Empty Set

We have mentioned some axioms, but have yet to show explicitly that any set at all exists. We wish to fix that here by constructing the empty set. This is usually accomplished in one of three ways1.

The first approach is to introduce an empty set axiom. Very often in books on set theory2 the existence of the empty set is given as an axiom with the note that said axiom is redundant, and used for emphasis.

The next two methods both make use of the axiom of specification. Let us assume for sake of argument that their exists some set A. Then by applying specification to A with the formula x \not = x, we obtain

\exists B\forall x(x \in B \iff (x \in A \land x \not = x)).

Since x \not = x is clearly false for all x, we have that the set B contains no elements. Thus we have that the existence of any set at all implies the existence of the empty set. The only remaining difficulty is proving that some set exists.

The second approach uses the axiom of infinity. Since the axiom of infinity asserts that there does exist a set (which is infinite) we can apply specification to obtain the empty set. This method feels very excessive as a means of postulating that some set exists, as we do so by asserting that their exist infinitely many. Additionally, this poses a problem in presentations like ours which add axioms as they become needed, as we wish to delay the introduction of the axiom of infinity until we examine infinite sets. This leads us to the third approach.

The final way to construct the empty set, and the one taken on this website, is to simply take as a logical axiom that our domain of discourse is not empty, that is \exists x (x = x). Recall the proof given earlier. There are many different formalizations of first order logic, so when working in a logic that allows empty domains of discourse one must necessarily use a different approach.

We may now begin to develop some properties of the empty set!

Properties

Footnotes

  1. The following paragraphs follow closely from the answer given by Andrés Caicedo on this [MathExchange] form.↩︎

  2. Enderton, Kunen, Weese & Just↩︎