Bertrand Russell is one of the giants of modern logic. His work in attempting to clarify the underpinnings of mathematics was unprecedented and comprehensive. Russell was interested in mathematics and systems of logic from a young age. In his mind math was the one thing that produced absolutely certain answers in life, an antidote to the insanity that haunted his family line.
Unfortunately, it quickly became obvious that math was not a perfect system. Math relied in many cases on “Axioms”, things that were just true because they had to be true. Here’s one that bugged Russell as a child learning geometry: if I draw a line, and then I draw a point outside of that line, only one line can be drawn that is parallel to the original and also crosses through this point. It’s a useful principle for geometric thinking, but where is the proof? Russell’s tutor could give him no reason for the fact beyond “it is assumed”. Russell became obsessed with eliminating axioms and building mathematics on a solid foundation of absolute provable fact, and worked with this idea all through his education. In fact, this quest for truth dogged him most of his adult life.
Russell for a time thought he was the only person driven to recreate math, but he soon found like-minded contemporaries and even a few mentors. The three that are most important to understand are Georg Cantor, David Hilbert, and Gottlob Frege. Cantor created Set Theory, and then went mad. Russell hoped these were unrelated. Frege created the Begriffsschrift, a language for logical operations as daunting as its crazy German name that was designed to examine the underpinnings of mathematics and build them on a solid foundation. The Begriffsschrift was based on Cantor’s Set Theory. Hilbert set out the challenge to mathematicians at the International Congress of Mathematics that Arithmetic, as it underpins all other maths, must be built on total certainty in order for mathematics to become impregnable to doubt. Hilbert, like Russell, shared the dream of a mathematics where any problem stated rigorously could be absolutely and with complete clarity solved. Russell set out to achieve this dream by using the underpinning of Set Theory and a logical language not unlike that developed by Frege.
It was around this time that Russell accidentally undid the work of Cantor and Frege, and gave Hilbert a rather nasty shock. It all had to do with Set Theory. A Set is a collection of objects defined by a common property. For instance, “the Set of all Green Things” would include a blade of grass, a leaf, and the Incredible Hulk. Similarly, the number 3 could be defined as “the Set of all Sets with Three Elements”. Three hats, three cars, three dogs, all of these are included in the infinite set of things that can be defined by the number 3. Set theory was at the heart of logic, it was the method that Russell, Frege, and many others were hoping to use to create a truly unassailable underpinning for mathematics. Russell killed that dream with a stray thought: What about a “Set of all Sets that do not contain themselves”?