User:CircuitCraft

I am a high-school student with a strong (borderline obsessive) interest in mathematics of every kind.

My contributions will be sporadic and diverse, with little overall direction. There are still some specific theorems I would like to see on, and I will (probably) eventually add them.

I will often attempt to contribute to fields of which I am not very familiar, by extrapolating from existing pages on this website and checking out books from the Internet Archive to use as sources. If I make a mistake, please correct it; I can handle constructive criticism.

Computability theory
I have been working on building a more comprehensive theory of computation on. So far, I have built an equivalence proof from recursive sets to Turing machines, which I will extend to recursive relations using a trick I have planned. The reverse proof would be most easily done with Minsky machines, after I prove Symbol Reduction (we can reduce the number of symbols on a Turing machine by increasing the number of states). The emulation of a Minsky machine with a URM is trivial.

I've also started working with provability, but I need to go back and change all of by Robinson Axioms]-based theorems to use [[Definition:Minimal Arithmetic|Minimal Arithmetic instead. I will also need to construct a recursive proof-finder. From there, Godel's First Incompleteness Theorem should be in reach.

As a long-term goal, I want to formally construct a "practical" programming language, to help with algorithm design. It should be usable for computational complexity, so this will need to happen after we define Random Access Machines (which are crucial for fine-grained complexity). The paper that defines RAMs uses a language they call "RAM-ALGOL," a subset of ALGOL that immediately translates into a RAM. We could also use LISP, as the original version has a simple structure and a lot of discussion about the implementation.

Non-Active Projects

 * Von Neumann-Bernays-Gödel Set Theory is Conservative Extension of Zermelo-Fraenkel Set Theory

Completed Projects

 * Equivalence of Definitions of Riemann and Darboux Integrals
 * Multitape Turing Machine Reduces to Turing Machine