Research Projects in the Foundations of Mathematics

Here are some ideas for research projects in the foundations of mathematics:

1. Sets are not the only data structure which could serve as a substrate for the axiomatic foundations of mathematics. One possible alternative is binary trees. Define isomorphisms between binary trees and all major mathematical concepts, and create a first-order theory of binary trees which can prove most of mathematics.
2. Axiomatize statistics in ZFC.
3. Construct a super-Turing machine by using real numbers. Define a variant on a Turing machine which uses real values instead of symbols, and functions over the real line instead of strings. Figure out what its properties are.
4. It seems intuitive that all of logic should be reducible to first-order logic. For instance, the set of first-order theories is isomorphic to the set of second-order theories. The only objection I have come across to the idea that second-order logic is not reducible to first-order logic is as follows. By the Lowenheim-Skolem theorem, every first-order theory has more than one model, whereas this has not been proven to be true of second-order theories. Do any important second-order theories have only one model? Is it a problem that first-order theories have more than one model? And, finally, is second-order logic reducible to first-order logic?
5. Category theory resists being formalized in axiomatic set theory, because of its need to talk about things like “the category of all sets.” Figure out how to formalize category theory.