My approach is dominantly model-theoretic and has focused on fragments of ZF, ZF with large cardinals, Quine-Jensen set theory NFU, and arithmetical systems of various flavors, ranging from fragments of PA, all the way to second order arithmetic and its subsystems.
One of my research projects deals with the comparative study of automorphisms of models of a variety of theories, ranging in strength from fragments of Peano arithmetic, all the way up to systems of set theory with large cardinals. The picture that has emerged from this work reveals that many foundational theories T (such as Peano arithmetic, second order arithmetic, and certain extensions of set theory with large cardinals) are characterized by the behavior of the automorphisms of models of T. This work also sheds light on the model theory of the Quine-Jensen system NFU of set theory with a universal set. I have completed four papers concerning automorphisms of models of foundational theories, available below, and I am working on-and-off on several others.
More recently, I have also started working on calibrating the interpretability strength of various foundational systems (including finite set theory, and theories with built-in satisfaction classes). My joint paper with Albert Visser and James Schmerl (available below) is the first paper in a projected series of papers dealing with this topic.
My latest work involves a new look at self-embeddings of models of arithmetic and set theory. Some of this work is in collaboration with Volodya Shavrukov.
I serve as an associate editor of the Bulletin of the Iranian Mathematical Society, dealing with papers in the areas of mathematical logic and set theory. You can find the latest issue of the Bulletin of the Iranian Mathematical Society online at http://bims.ims.ir .
Recent Papers :
New Constructions of full satisfaction classes (with Albert Visser).
Also, the paper below is ready; send me an e-mail if you are interested in obtaining a copy:
A New Proof of Tanaka’s Theorem (every nonstandard countable model of WKL0 has a nontrivial self-embedding onto an initial segment of itself).
Automorphisms of models of bounded arithmetic, Fundamenta Mathematicae, vol.192 (2006), pp. 37-65.
From bounded arithmetic to second order arithmetic via automorphisms, in Logic in Tehran, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, 2006.
Omega-models of finite set theory [with James Schmerl and Albert Visser], in Set theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (edited by J. Kennedy and R. Kossak), Cambridge University Press, 2011.
An improper arithmetically closed Borel subalgebra of P(omega) mod FIN [with Saharon Shelah], Topology and Its Applications, December 2011, pp. 2495-2502, journal copy available here.
Minimal elementary extensions of models of set theory and arithmetic, Archive for Mathematical Logic, (1990), no. 3, 181-192.
Undefinable classes and definable elements in models of set theory and arithmetic. Proceedings of American Mathematical Society, 103 (1988), no. 4, 1216-1220
Counting models of set theory. Fundamenta Mathematicae vol. 174 (2002), no. 1, pp. 23-47.
Power-like models of set theory, Journal of Symbolic Logic (2001), no. 4, pp. 1766-1782.
Analogues of the MacDowell-Specker theorem for set theory. Models, algebras, and proofs (Bogotá, 1995), pp. 25-50, Lecture Notes in Pure and Appl. Math., 203, Dekker, New York, 1999.
Conservative extensions of models of set theory and generalizations.,J. Symbolic Logic, vol. 51 (1986), no. 4, pp. 1005-1021.
Weakly compact cardinals in models of set theory, Journal of Symbolic Logic, vol. 50 (1985), no. 2, pp. 476-486.
On certain elementary extensions of models of set theory. Transactions of American Mathematical Society, vol. 283 (1984), no. 2, pp. 705-715.
Model Theory of the Regularity and Reflection Schemes [with Shahram Mohsenipour], Archive for Mathematical Logic, Vol. 47, #5 (2008), pp. 447-464.
Trees and Keisler's problem, Archive for Mathematical Logic, 40 (2001), no. 4, pp. 273-276.
Delta as a continuous function of x and epsilon. American Mathematical Monthly, 107 (2000), no. 2, pp. 151-155.
Nonmetrizability of uncountable well-ordered spaces [with A. Abian], Simon Stevin (published in Belgium) 55 (1981), no. 1-2, pp. 3--6.
Logic in Tehran, Proceedings of the Logic, Algebra, and Arithmetic conference held in Tehran during October 2003, edited by A. Enayat, I. Kalantari, and M. Moniri, Lecture Notes in Logic Series, vol. 26, Association for Symbolic Logic, La Jolla, CA; A K Peters, Ltd., Wellesley, MA, 2006.
Nonstandard Models of Arithmetic and Set Theory, Contemporary Mathematics, vol. 361, American Mathematical Society (2004), edited by A. Enayat and R. Kossak, American Mathematical Society, Providence, RI, 2004.
University of Oslo Logic Seminar (Norway, Sep 13, 2012)
University of Gothenburg Logic Seminar (Sweden, Sep 14, 2012)
Pure Mathematics Seminar, University of East Anglia (May, 2010, Norwich, England)
Model Theory Seminar, University of Leeds (May 2010, Leeds, England)
Logic Seminar, University of Manchester (May 2010, Manchester, England)
Oxford Logic Seminar (April 2007, Oxford, England)
NYC Logic Conference (May 2005, New York City)
Logic, Algebra, and Arithmetic (October 2003, Tehran, Iran)
· RECENT CONFERENCE ORGANIZATION
ASL Winter Meeting (with Joint Mathematics Meetings) of the Association for Symbolic Logic (January 2009, Washington, DC)
Logic Colloquium 2008 (July 2008, Bern, Switzerland)
· GRADUATE STUDENTS
Betsy Andersen (project topic: Coding Theory), completed Spring 1994.
Omar Mirza (project topic: Gödel’s Theorem), completed Summer 1994.
Blair Jones, (thesis topic: Ramsey Theory), completed Spring 1995.
Michelle Perschbacher (project topic: Music and Number Theory), completed Summer 1997.
Valbona Bejleri (project topic: The Probablistic Method), completed Summer 2001.
Adeniran Adeboye (project topic: Combinatorial Number Theory), completed Spring 2002.
Anna Rose Haralampus (project topic: Fractals and Topology), completed Spring 2003.
Caleb Rossiter (thesis topic: Relativity Theory), completed Spring 2004.
Stephen Wheatley (project topic: Nonstandard Analysis), completed Spring 2006.
Mahmoud Momenipour IASBS (thesis topic: Logical Foundations of Nonstandard Analysis), completed Fall 2006.
Kun Zhao, (project topic: Laws of Large Numbers), completed Summer 2010.
Adam Moskey (thesis topic: Continuous Selections), completed Fall 2010.
Michael Headley (project topic: Transcendental Numbers), completed Fall 2010.
Michael Cassel (project topic: Automorphisms of Models of PA), completed, Spring 2013.
Amanda Purcell (thesis topic: Ultraproducts and Applications), in progress.