- Department of Mathematics
Bromfield-Pearson, Room 105
Set Theory, specifically forcing elementary embeddings and large cardinal axioms
In 1930 Gödel proved his famous Incompleteness Theorem showing that any reasonable system in which to do mathematics is incapable of answering all mathematical questions. Because of this insight the process of discovering, cataloging and comparing independent statements has become an active area of research. I study statements which can not be proven true or false from a given axiom system.
Some of the questions to I try to answer about such statements are:
- Is that statement relatively consistent with ZFC (and can we prove it is?)
- What is the consistency strength of the statement (i.e what additional axioms must we add to ZFC in order to prove it is relatively consistent?)
- What does it imply if assumed to be true?
- What other undecidable statements are consistent/inconsistent with the statement?
- Most recently I have been examining the consistency strength of stationary set reflection at successors of singular cardinals, for instance ℵω+1ℵω+1. My research makes heavy use of large cardinal properties, elementary embeddings and forcing.