Bertrand Russell (1872-1970)
It isn’t too often that we think about or even explore the fundamental building blocks of mathematics. These building blocks are called axioms. Axioms are statements taken to be true, i.e. they cannot be proven. This causes mathematicians, whether they know it or not, to take a lot of things on faith. All of modern mathematics is built on top of axiomatic systems, which is a list of axioms from which one can logically derive theorems. We further require that no axiom on the list can be proven by using other axioms. In this series we will look at some interesting axioms and axiomatic systems that govern mathematics. We will begin with perhaps one of the most important systems and that is sets. One could argue that the system of logic used is the most important. We are assuming a working knowledge of tools used in logic, and will forgo formalizing such a thing. We often think about an axiomatic system like a tree.
But first let us get a feel for why an axiomatic system is so important. This brings us to something called naive set theory. This naive set theory is an example of common occurrences in mathematics in which we define things by using natural language. We often wind up tossing words around until someone formalizes an idea. This happened with set theory.
DEFINITION: The collection of all things satisfying a property forms a set. In symbols, if is the statement “ has property “, then a set can be written in the form .
From this definition one can deduce that the collection of all things that are sets is a set. In symbols, if is the statement is a set, then is a set. But this means contains itself. Now that we know one exists, let us call all sets that contain themselves abnormal. In symbols, a set is abnormal if . Where means is an element of.
Obviously does not contain itself as a set. We will call these sets normal. In symbols, a set is normal if . It is clear that these two definitions are exclusionary and all sets are either normal or abnormal. Now consider the following set.
This is simply the collection of all normal sets. By definition, this is indeed a set. But now it must either be normal or abnormal. If is normal, then . By definition of we have which is a contradiction.
On the other hand if is abnormal then , hence by definition of we have $R\notin R$ another contradiction. We conclude that is neither normal nor abnormal. Which is our ultimate contradiction which tells us that naive set theory is inconsistent.
This example is known as Russell’s Paradox (1901) and it has many equivalents in naive set theory). Even though naive set theory has this major flaw, it is still very useful in teaching and introducing students to the real heart of mathematics. In the next post we will explore the Zermelo-Fraenkel (ZF) axiomatic system which is the most widely used system in mathematics today. We will also make note of ZFC which includes the historically controversial Axiom of Choice.