Ali Enayat
RESEARCH
I am a mathematical logician, with a strong interest in the metamathematics of foundational axiomatic systems such as Zermelo-Fraenkel set theory (ZF) and Peano arithmetic (PA).
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 WKL_{0} 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.
Proceedings of the IPM Logic Conference 2007, Special Issue of the Annals of Pure and Applied Logic, (guest) edited by A. Enayat and I. Kalantari, March 2010.
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)
Young Set Theory Workshop, (March 2011, Königswinter, Germany), slides
1, slides
2
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)
Kunen Fest: Topology and Set Theory Conference
(April 2009, University of Wisconsin, Madison), slides
IPM
Logic Conference 2007 (June 2007, Tehran, Iran), slides(1), slides(2)
University of Paris Logic Seminar (May 2007, Paris,
France), slides
70^{th} Anniversary of NF (May 2007,
Cambridge, England), slides
University of Manchester Logic Seminar
(April 2007, Manchester, England), slides
Oxford Logic Seminar (April 2007, Oxford,
England)
Amsterdam-Utrecht Logic Colloquium (April 2007,
Utrecht, The Netherlands), slides
NYC
Logic Conference (May 2005, New York City)
Logic, Algebra, and Arithmetic (October 2003, Tehran, Iran)
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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)
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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.