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Chapter 3 of "Naive Set Theory"

Let's take a look at the so-called "empty set." An empty set is a set without any elements. We had previously introduced sets as collections of things called members or elements. An empty set is a set with...well...nothing. No elements. Applying the Axiom of Specification which we discussed in our previous article to the sentece x ≠ x results in the set {x∈A: x ≠ x}. x ≠ x, of course, is a false sentence. It violates the law of identity, according to which everything is identical to itself.

Halmos points out that the Axiom of Extension implies that there can be one such set. This set is called the "empty set", as we have seen. The empty set is symbolized by Ø. Ø ⊂ A for every A (or, for that matter, any other set symbolized by a letter). Halmos notes that we must prove that every element in Ø (although it has no elements) belongs to A (or, again, any other set).

Now, how do we prove that it is false that Ø ⊂ A? One could only demonstrate such a thing if Ø possessed an element that was not a member of A. But Ø does not possess such an element. Indeed, it does not possess any element! We therefore prove by reductio ad absurdum that Ø ⊂ A. Since it is not false, it must necessarily be true that Ø ⊂ A.

Let's look at an Axiom called the Axiom of Pairing.

Axiom of Pairing (or Axiom of Pairs) - For any two sets, there is a set to which both sets belong.

To put it a little more formally, if there are sets a and b, A is such that a∈A and b∈A. Halmos notes that a consequence of this, as well as another way of formulating the Axiom of Pairing, is to note that there is a set that not only contains both sets, but contains literally nothing else. Let's look at a strictly formal articulation of this axiom:

Axiom of Pairing - ∀x∀y∃z∀w(w∈z ≡ w=x ∨ w=y)

In other words, every w is a member of at least one z if and only if w is equal to x, or w is equal to y.

If we apply the axiom of specification to the set A, along with the formula "x = a or x = b" then we get

{x ∈ A : x = a or x = b}

This set contains only sets a and b. Halmos notes that it is because of the axiom of extension that only one set can possess this property. This sort is ordinarily symbolized by {a, b}. Even those with little education in mathematics will recognize this as a "pair." The lone set "a" can be used to form the unordered pair {a, a}. Since both elements of the ordered pair are identical, the ordered pair only has one element, and is referred to as a "singleton." It can be signified by {a}.

Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.

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