As one peers at a radiating metropolis of buildings piercing the night sky, it is easy to imagine the thought and engineering that produced such a modern wonder. The intricate detail and unique designs appeal to the artistic senses. Before each building’s architectural beauty is manifested, a strong foundation must be poured. Such a concept is inherent throughout physical law and abstract mathematics.
In the trenches of abstract mathematics set theory forms its foundations. Georg Cantor defined a set as, “a gathering together into a whole of definite, distinct objects of our perception or of our thought- which are called elements of the set.” This definition is known as a naïve approach to set theory and has (as will be shortly described) cracks. One such man to find these cracks was Bertrand Russell.
In naïve set theory, the idea that any set can be formed without contradiction is an illusion. Russell showed this with the following Paradox:
Russell’s Paradox (1901): Suppose there exists a set A such that the only elements of A are the set of all sets that do not contain themselves. Does A contain itself?
One can see the apparent contradiction that arises as a consequence of the naïve set approach. If A contains itself, then it is in contradiction with the description that A only contains sets that do not contain themselves. On the contrary, if A does not contain itself, then this is also a contradiction since A contains all of the sets that do not contain themselves.
Another version derived from Russell’s Paradox is the Barber Paradox. Basically this paradox proposes that, if there exists only one Barber in a small town that exclusively shaves men that don’t shave themselves, then who shaves the Barber? The same self-contradiction emerges as with the original version of Russell’s Paradox.
These self-contradictions required the development of a more formal set theory. This was accomplished by two mathematicians, Ernst Zermelo (1908) and Abraham Fraenkel (1921), who developed an axiomatic approach to set theory. Zermelo-Fraenkel set theory avoids the contradiction of Russell by ruling it out using defining axioms.
Set theorists in logic and mathematics continue probing the foundations looking for new cracks to patch. Some of the cracks are not easily visible and remain hidden waiting for the right approach to expose them. Lurking behind the shadows, you may find one waiting to be discoverd. What do you think about the foundations of mathematics and set theory?