On the Method of Arithmetization and its Applications
A central topic in both logic and philosophy are paradoxes such as the Liar, Russell's paradox, and the sorites.
A major goal of Dr Walter Dean's Fellowship is to determine if such puzzles can be resolved uniformly by an analogy to mathematical undecidability results in which it is shown that a statement is neither provable nor refutable from given axioms - e.g. the independence of the Parallel Postulate from the axioms of Euclidean geometry or the Continuum Hypothesis from the axioms of Zermelo-Fraenkel set theory.
Dr Dean will locate this question relative to developments in the foundations of mathematics which subsume Hilbert's program during the 1900s-1930s -- i.e. that of axiomatizing infinitary mathematics so that it can be proven to be consistent using finitary methods alone -- and as well as subsequent results in mathematical logic which it inspired.
He will focus on applications of a technique known as the method of arithmetization which grew out of Hilbert's Foundations of Geometry and is embodied in Bernays's formalization of Gödel's Completeness Theorem in first-order arithmetic. Additional questions he will investigate include the following: how does the metamathematical investigation of the Completeness Theorem bear on the question of whether consistency implies existence? How do results about interpretability bear on the relationship between formal independence in arithmetic, geometry, and set theory? Are there absolutely undecidable number theoretic statements?