Eric L. Nichols

Research and Set Theory

When I see something new I want to learn how it works. My general path to answering these questions has always been to disassemble it and put back together. This works great for physical objects but what about more abstract ideas such as, "how do you solve a Rubik's cube?".  This quickly devolves into deep logic questions like "How do you disassemble and algorithm?" and "How do you know an algorithm works?".

To me Set Theory represents the "disassemble it and put it back together" mentality for answering these more abstract mathematical questions. A Set Theorists answer do our Rubik's cube question is to encode the state of the Rubik's cube as a set, and algorithms of functions between the encoding set. We can then use the properties of sets to answer all the questions we have. For every problem we disassemble it into sets (foundational pieces) and put it back together, using the properties of sets. This same ideology for problem solving shows up heavily in all flocks of science. Where instead of sets of we code problems into binary (computer science), elements (chemistry), or sub-atomic particles (physics).

Within set theory we can ask answer questions about the structure of mathematics itself, not just a specific object or pattern. I am particularly enthralled with questions about the existence of extremely large cardinals. Cardinals where, if they exist in mathematics, their properties create interesting structures and answers for questions whose consequences range across all sorts of different branches of mathematics and science in general.