PhD Research

Recent PhD's
My recent PhD students have completed research into topics broadly speaking in Inner Model Theory which seeks to generalise Gödel's universe of constructible sets. Typically this involves interactions with medium size cardinals, or with determinacy at low levels of the projective hierarchy.

Determinacy of two person perfect information games. The interaction between the assumption of winning strategies for infinite games played on integers into subsets of Baire space and inner models of ZF set Theory is well studied.
Some extensions of determinacy in the difference hierarchy of co-analytic sets, Chris Le Sueur.
See also:
Determinacy of refinements to the difference hierarchy of co-analytic sets, by Chris Le Sueur, Annals of Pure and Applied Logic 169 n1 (2018 01 01): 83-115.

The combinatorics of the Gödel constructible universe. Generalising the notions of closed unbounded, and stationary sets, Hazel Brickhill.
This thesis, besides generalising the notion mentioned into a transfinite hierarchy, also defined higher generalisations of Jensen's Square principle, and showed these hold precisely where certain categories of indiscernibility fail at weakly compact cardinals.

Ramsey cardinals and long games Ramsey cardinals are a medium sized cardinal weaker than measurable cardinals but defined through certain indiscernibility properties.
Virtual Set Theory, Dan Nielsen.
Here long games were played on structures in the universe that used small large cardinal methods (of Measurable and Ramsey cardinals and below) to work out strategies for such games, and conversely in some cases. Another part concerned forcing methods to get the consistency strength of smaller large cardinals (consistent with V=L) which 'virtually' reflect the properties of much stronger `real' large cardinals in V, but inconsistent with L.
See also: Games and Ramsey-like cardinals with D. Nielsen, in Journal of Symbolic Logic, 84 No. 1, 2019, 408-437.

I am interested in

Higher Type Recursion Theory and Transfinite Machine Models. Such models interact with levels of determinacy in the arithmetic hierarchy (at the G_delta,\sigma level)

Higher Type Recursion for Transfinite Machine Theory in Proceedings of Computation in Europe, CiE2019, Eds. F. Manea, B. Martin, D. Paulusma and G. Primiero, Springer Lecture Notes in Computer Science, 11558, 72-83.

G_delta,sigma Games in the Isaac Newton Institute Pre-print series No. NI12050-SAS, July 2012.

I have also written on the Global Reflection Principle:

Reflecting on Absolute Infinity, with Leon Horsten, in Journal of Philosophy, 113, Feb. 2016, 89-111.

and

Obtaining Woodin's Cardinals in ``Logic in Harvard: Conference celebrating the birthday of Hugh Woodin'' Eds. A. Caicedo, J. Cummings, P.Koellner & P. Larson, AMS Series, Contemporary Mathematics, vol. 690, 161-176, May 2017.

and Theories of Truth, see for example:

Some observations on truth hierarchies in the Review of Symbolic Logic, 7 No. 1, March 2014 1-30.