The Prospects for Mathematical Logic in the Twenty-First Century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

[1]  Joaquín Pascual,et al.  Infinite Abelian Groups , 1970 .

[2]  W. Woodin,et al.  Supercompact cardinals, sets of reals, and weakly homogeneous trees. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[3]  László Fuchs,et al.  Infinite Abelian groups , 1970 .

[4]  André Cécyre On the eve of the year 2000 , 1997 .

[5]  R. Stephenson A and V , 1962, The British journal of ophthalmology.

[6]  A. Kechris Classical descriptive set theory , 1987 .

[7]  John R. Steel,et al.  A proof of projective determinacy , 1989 .

[8]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[9]  Thomas Jech Martin's Maximum , 1987 .

[10]  Marcia J. Groszek,et al.  A basis theorem for perfect sets , 1998, Bull. Symb. Log..

[11]  Saharon Shelah,et al.  Simple unstable theories , 1980 .

[12]  Chi Tat Chong,et al.  The polynomial topological complexity of Fatou-Julia sets , 1995, Adv. Comput. Math..

[13]  S. Barry Cooper On a Conjecture of Kleene and Post , 2001, Math. Log. Q..

[14]  Shmuel Weinberger,et al.  The Fractal Nature of Riem/Diff I , 2003 .

[15]  Arkadii M. Slinko,et al.  Degree spectra and computable dimensions in algebraic structures , 2002, Ann. Pure Appl. Log..

[16]  Chi Tat Chong Positive Reducibility of the Interior of Filled Julia Sets , 1994, J. Complex..

[17]  S. Shelah,et al.  Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II , 1988 .

[18]  Manuel Lerman,et al.  Computability Theory and Its Applications , 2000 .

[19]  Harvey M. Friedman,et al.  A Borel reductibility theory for classes of countable structures , 1989, Journal of Symbolic Logic.

[20]  A. Kechris,et al.  The Descriptive Set Theory of Polish Group Actions: BETTER TOPOLOGIES , 1996 .

[21]  Greg Hjorth Around nonclassifiability for countable torsion free abelian groups , 1999 .

[22]  S. Cooper,et al.  The jump is definable in the structure of the degrees of unsolvability , 1990 .

[23]  H. S. Allen The Quantum Theory , 1928, Nature.

[24]  H. Keisler,et al.  The Kleene Symposium , 1980 .

[25]  Alain Louveau,et al.  Descriptive Set Theory and the Structure of Sets of Uniqueness , 1987 .

[26]  Richard A. Shore,et al.  Models and Computability: Effective Model Theory: The Number of Models and Their Complexity , 1999 .

[27]  Anand Pillay,et al.  Simple Theories , 1997, Ann. Pure Appl. Log..

[28]  H. Jerome Keisler Quantifier Elimination for Neocompact Sets , 1998, J. Symb. Log..

[29]  David Deutsch,et al.  Machines, logic and quantum physics , 2000, Bull. Symb. Log..

[30]  Greg Hjorth,et al.  Classification and Orbit Equivalence Relations , 1999 .

[31]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[32]  Theodore A. Slaman On a question of Sierpiński , 1999 .

[33]  R. Jensen,et al.  The fine structure of the constructible hierarchy , 1972 .

[34]  Richard A. Shore Computable Structures: Presentations Matter , 1999 .

[35]  Stevo Todorcevic,et al.  Partition Problems In Topology , 1989 .

[36]  A. Turing On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[37]  Patrick Dehornoy,et al.  Braids and self-distributivity , 2000 .

[38]  W. Hugh Woodin,et al.  The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , 1999 .

[39]  Neil Immerman,et al.  On the Unusual Effectiveness of Logic in Computer Science , 2001, Bulletin of Symbolic Logic.

[40]  Richard A. Shore,et al.  Defining the Turing Jump , 1999 .

[41]  A. Kechris Actions of Polish Groups and Classification Problems , 2003 .

[42]  A. Troelstra Models and computability , 1973 .

[43]  D. Deutsch Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[44]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[45]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[46]  Alain Louveau,et al.  Descriptive set theory and dynamical systems , 2000 .

[47]  Ehud Hrushovski,et al.  Zariski Geometries , 1993 .

[48]  R. Camerlo,et al.  The completeness of the isomorphism relation for countable Boolean algebras , 2000 .

[49]  Ehud Hrushovski,et al.  The Mordell-Lang conjecture for function fields , 1996 .

[50]  Harvey M. Friedman,et al.  Finite functions and the necessary use of large cardinals , 1998, math/9811187.

[51]  Matthew Foreman Review: Donald A. Martin, John R. Steel, Projective Determinacy; W. Hugh Woodin, Supercompact Cardinals, Sets of Reals, and Weakly Homogeneous Trees; Donald A. Martin, John R. Steel, A Proof of Projective Determinacy , 1992 .

[52]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[53]  Gianluigi Oliveri,et al.  Truth in Mathematics , 1998 .

[54]  H. G. Dales,et al.  An Introduction to Independence for Analysts , 1987 .

[55]  Ehud Hrushovski,et al.  Finite Structures with Few Types , 1993 .

[56]  J. Kahane,et al.  Ensembles parfaits et séries trigonométriques , 1963 .

[57]  Richard A. Shore,et al.  A splitting theorem for n - REA degrees , 2001 .

[58]  A. Wilkie Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , 1996 .

[59]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[60]  Richard A. Shore,et al.  Natural Definability in Degree Structures , 1999 .

[61]  Robin Gandy,et al.  Church's Thesis and Principles for Mechanisms , 1980 .

[62]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[63]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[64]  Benedikt Löwe,et al.  An introduction to core model theory , 1999 .

[65]  Saharon Shelah,et al.  Infinite abelian groups, whitehead problem and some constructions , 1974 .

[66]  Theodore A. Slaman Aspects of the Turing Jump , 2002 .

[67]  Byunghan Kim Forking in Simple Unstable Theories , 1998 .

[68]  Y. Moschovakis Descriptive Set Theory , 1980 .

[69]  Ya'acov Peterzil,et al.  A trichotomy theorem for o‐minimal structures , 1998 .

[70]  Richard A. Shore,et al.  Characterizations for computable structures , 2000 .