Here are some ideas for research projects in the foundations of mathematics:
- 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.
- Axiomatize statistics in ZFC.
- 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.
- 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?
- 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.