MTH 217: Mathematical Structures

In pure mathematics, our business is establishing true facts about the mathematical universe. The test of truth is proof. If we can prove a fact, we call it a theorem. You may have noticed, however, that in proving theorems, we use theorems we have previously proved. To prove those theorems, we must use still earlier theorems and so on and so on. How does it all begin?

Mathematical Structures is about how it all begins.

The actual content of Mathematical Structures varies each year it is taught, but the key elements are those ideas and constructs that are fundamental to all mathematics. They include the axiomatic method, sets, the construction of number systems, and Cantor's theory of the infinite.

The axiomatic method. The typical high school geometry course starts with a number of postulates or axioms, statements that are accepted without proof as the basis for all that follows. The same is true for most branches of mathematics. In number theory, there are axioms for arithmetic. In analysis, there are axioms for the real numbers. In algebra, there are axioms for groups, rings, and fields. In each case, certain concepts are accepted as intuitively understood, and certain facts about them are assumed as intuitively true.

This is the ``axiomatic method.'' It is the framework of theoretical mathematics. The most delicate and difficult part of it is proof, and Mathematical Structures generally devotes a fair amount of time to developing both its skill and its art.

Sets. The concept of a set is one of the most primitive in mathematics. We think of a set as any collection of elements: the set of all students at Smith, the set of all natural numbers, the set of all straight lines in the plane, the set consisting of: -2.17, $\pi$, and Sophia Smith. Sets are important, useful, and simple.

They can also be trivial (the empty set), useless (the set of all three-legged tables in Dade County, Florida), and complex (see below). The point is that they are the clay from which anything mathematical can be fashioned.

Constructing number systems. The importance and usefulness of sets is demonstrated by showing that all our number systems can be built from sets. Consider, for example, the natural numbers: 0, $1$, $2$, $3$,.... We can choose the empty set, $\emptyset$ or $\{\}$, to represent 0. For $1$, we can choose $\{\emptyset\}$, a set with exactly one element. For $2$, we can choose a set with two elements: $\{\emptyset, \{\emptyset\}\}$ or $\{0, 1\}$. If we keep going, we get $3 = \{0, 1, 2\} = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$, and so on.

A much more sophisticated construction is required to build the real numbers. It was first accomplished in the nineteenth century and was a milestone in mathematics. The building blocks are not simple. They are infinite sets of rational numbers.

Cantor's theory of the infinite. Infinity is perhaps the most romantic idea in mathematics. It combines analysis with philosophy and theology. It is mysterious and dangerous. Aristotle denied its absolute existence. It enraptured Bruno. It frightened Pascal.

Galileo played with infinity and noticed that two infinite sets:

$$1, 2, 3, 4, 5, 6,\ldots$$   and$$\qquad 1, 4, 9, 16, 25, 36,\ldots$$

could be paired up, that is, we can match them element for element:

$$\begin{array}{ccccccc} 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ \u... ...& \hphantom{00} & \hphantom{00} & \hphantom{00} & \end{array}$$

This would suggest that the two sets are the same size. Galileo thought that was impossible, since the second set is contained in the first. He concluded that with infinite sets, the concept of ``size'' makes no sense. It wasn't until the nineteenth century that Georg Cantor discovered that it can make sense, and more surprisingly, that there are infinite sets of different sizes. Cantor's theory is among the most provocative, the most magical, in mathematics.

Finally, a complex set.

One would expect that if we had a property, like ``being an even number,'' or ``being retired high school teacher,'' could form the set of all objects with that property, that is, there is a set consisting of all even numbers, and there is a set consisting of all retired high school teachers.

Consider, however, this property: ``being a set which is not a member of itself.'' Let $R$ be the set of all sets which are not members of themselves. There are some sets we know are in $R$. The set of all even numbers, for example, since that set is not itself an even number. There are some sets that would not be in $R$, for example, the set of all sets must be a member of itself. There is a problem with $R$, however. The problem is this: Is $R$ a member of $R$?