empty set

An empty set is the set, denoted by Ø or {}, that has no members; also known as the null set. This is not the same as zero, which is the number of members of Ø. Nor is Ø the same as nothing because a set with nothing in it is still a set, and a set is something. The empty set, for example, is the set of all triangles with four sides, the set of all numbers that are bigger than 9 but smaller than 8, and the set of all opening moves in chess that involve a king. Applying the concept of the empty set helps distinguish between the different ways that "nothing" is used in everyday language. In his book What Is the Name of This Book? (1978),1 Raymond Smullyan wrote:


Which is better, eternal happiness or a ham sandwich? It would appear that eternal happiness is better, but this is really not so! After all, nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore a ham sandwich is better than eternal happiness.


What is wrong with this? The first statement is equivalent to "The set of things that are better than eternal happiness is Ø." The second statement is equivalent to "The set {ham sandwich} is better than the set Ø." The confusion arises because the first is comparing individual things, while the second is comparing sets of things, and Ø plays a different role in each.



1. Smullyan, Raymond M. What is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles. Englewood Cliffs, NJ: Prentice Hall, 1978.