Lesson 3 in Perceptive Mathematics: singletons

A singleton is a set having exactly one element. For instance, {R} is a singleton, where R is a rose. Now, we do want to think of a rose R as a different object to the singleton {R}. These objects even have different types. The first is a plant, and the second is a set. Here is an illustrative analogy: a rose is not the same thing as as a vase with a rose in it.

The empty set {} is not a singleton, since it does not have any elements. Neither is a set having two or more different elements a singleton. We can turn any object B into a singleton, by considering the set having B as its unique element, i.e., the set {B}. What happens when B={}? In notation, we get {B}={{}}. Now, what is the set {{}}? Is it still the empty set? Let us see. The empty set does not have any elements. What about the set {{}}? Does it have any elements? Well, this set is a singleton (remember, it is equal to {B}, where B={}), so yes, it has exactly one element, that element being the empty set B={}. So, since {} has no elements and {{}} has an element, these two sets cannot be the same set (if they were the same set, they would have the same elements).

Exploration 3.1. Decide whether the set {{{}}} is the same set as the set {{}}.

Consider again the set {R}. Is there a way to partition this set? Recall that during partitioning we arrange elements of a set in classes. Since a class cannot be empty (the third axiom for partitioning), there is only one class that can be formed, the class consisting of the unique element of the given set, so it is the class {R}. What is the corresponding quotient? Well, a quotient is the set of all classes formed during partitioning. Since there is only one class that can be formed, we get that the quotient is the singleton {{R}} (elements of the partition must be the classes). So if V={R} then the singleton of V is its quotient {V}={{R}}, which happens to be the only quotient that V={R} has.

Note that the singleton {R} and its quotient {{R}} are different sets, since, they do have different elements. The element of the first singleton is R, the rose, while the element of the second singleton is the set {R}, and we already agreed that R and {R} are different from each other. 

Exploration 3.2. How many quotients does the empty set have? Is the quotient of the empty set {} the singleton {{}}? Revisit the axioms for partitioning to get help with answering this question.

Here is another way of understanding singletons and sets in general. The thought of a rose is different from the actual rose. The thought of the thought of a rose is different from both. Every time we add "the thought of" we can interpret this as a process of taking a singleton:

  • R is the actual rose
  • {R} is the thought of a rose
  • {{R}} is the thought of the thought of a rose, etc. 

In this analogy, the empty set can be interpreted as the thought of "nothing". Nothing does not exist, but the thought of nothing does.