What is a set?
Simply put, a set is a collection of objects or entities; these are called the elementsof
the set. The number of elements in a set can be large, small, finite, or infinite. The
informal notation used for sets is sometimes seen as x{y, z,...}, with brackets used
to contain the elements within the set. It is stated as, “x is a set consisting of the ele-
ments y, z, and so on.” But more commonly sets are seen as capital letters and ele-
ments as lowercase letters, such as ais an element of set A.
How does one interpret sets?
There are several ways to look at sets. Two sets (or more) are considered identical if,
and only if, they have the same collection of objects or entities. This is a principle
known as extensionality. For example, the set {a, b, c} is considered to be the same as
set {a, b, c}, of course, because the elements are the same; the set {a, b, c} and the set
{c, b, a} are also the same, even though they are written in a different order.
It becomes more complex when sets are elements of other sets, so it is important
to note the position of the brackets. For example, the set {{a, b}, c} is distinct from the
set {a, b, c} (note that the brackets differ); in turn, the set {a, b} is an element of the
set {{a, b}, c}. (It is a set included between the outside brackets.)
Another example that shows how sets are interpreted includes the following: If B
is the set of real numbers that are solutions of the equation x^2 9, then the set can be
written as B{x: x^2 9}, or Bis the set of all xsuch that x^2 9. Thus Bis {3, 3}. 123
FOUNDATIONS OF MATHEMATICS
Who developed set theory?
G
erman mathematician George (Georg) Ferdinand Ludwig Philipp Cantor
(1845– 1918) is most well known for his development of set theory (for more
information on Cantor, see “History of Mathematics”). His Mathematische
Annalenis a basic introduction to set theory in which he built a hierarchy of infi-
nite sets according to their cardinal numbers. In particular, using one-to-one pair-
ing, he showed that the set of real numbers has a higher cardinal number than
does the set of rational fractions.
Unlike most subjects in mathematics, Cantor’s set theory was his creation
alone. But like many brilliant, revolutionary thinkers throughout history, his
ideas were highly criticized by his contemporaries. This strong opposition con-
tributed to the multiple nervous breakdowns he suffered throughout the last 33
years of his life, which ended tragically in a mental institution.