This spring, I wrote a research paper defending Mathematical Platonism, or the view that numbers exist outside of the mind. Numbers are abstract entities that exist in some non-mental and non-physical realm that we access through reason, not perception. A large part of my argument relied on the reductionist approach that math could be reduced down to set theory, or a theory regarding sets of objects. This approach somewhat bridged the gap of one of the largest problems in Mathematical Epistemology: if numbers exist in an abstract realm, how do we perceive them?

I wrote a lengthy argument about it and discovered a roundabout solution called Plenitudinous Platonism. The view stated that all possible mathematical entities existed as sets, and therefore we only needed to create axioms to work with them. This was certainly a long-winded answer, but it provided an answer at least. Plenitudinous Platonism relied on set theory, and without that, it would be an incoherent argument.

I stumbled over Russell’s paradox a month after finishing the paper. Developed by logician Bertrand Russell, the paradox reveals an inconsistency within set theory. Russell was a logicist, and he had spent several years trying to prove that math reduced to set theory. The paradox relies on two rules of set theory, developed by Russell himself. I will explain them down below.

Set theory is the discipline regarding the relationship between objects in a—you guessed it—set. The first rule of naive set theory is called the unrestricted comprehension principle. It basically means that a set can be created with any possible condition. So a set can be created of all x where x is a prime number, or x is a multiple of 17, and so on. Based on this logic, you can extrapolate numerous rules, including that sets can contain themselves. This is just part of principle one. If you can make a set with any possible condition, that condition can be used to create a set of all x such that x is a set. The tricky part about this is what to do when considering a set of x such that x are sets that do not contain themselves. This reveals a logical contradiction. Before moving on, try to take some time and see if you can figure out the problem for yourself.

Here it is. If the set of all sets that do not contain themselves does contain itself, it does not contain itself. If it does not contain itself, then it does. This still may be pretty difficult to understand, so I will explain with a famous example derived from the original paradox. Consider a barber in town. This barber is an extremely generous man, so he cuts the hair of all and only the men in town who do not cut their own hair. Now consider, does the barber cut his own hair? If the barber does cut his own hair, he is not the barber specified in the question, since that barber only cuts the hair of men in town who do not cut their own hair. However, if the barber does not cut his own hair, he fits into the group of people the barber shaves, and therefore does cut his own hair. The barber both must and must not shave himself under the comprehension principle, meaning the paradox has no logical solution.

When Russell sent this paradox to his colleague, Gottlob Frege, Frege was sent into an extreme state of dismay. Russell eventually found a way to work around this problem—by removing the axiom of unrestricted comprehension. If not all sets could be made, such as sets that contain themselves, no contradiction can arise. Although modern Zermelo-Fraenkel set theory (ZFC) has addressed these issues with more rigorous axioms, it still feels to me like a patch rather than a completely satisfying foundation for mathematics. I will write part two once I have done more research.