# The Helsinki Logic Group

## Members

##### Members - Research - Publications - Links - Contact Info
Aapo Halko , Ph.D., descriptive set theory

Åsa Hirvonen, Ph.D., model theory of metric structures

Taneli Huuskonen , docent, model theory, set theory, logic and analysis

Tapani Hyttinen , docent, stability theory, infinitary logic

Daisuke Ikegami, Ph.D., postdoctoral researcher, set theory

Juliette Kennedy, university lecturer, philosophy of mathematics, set theoretic model theory.

Meeri Kesälä, Ph.D., model theory

Juha Kontinen, Ph.D., finite model theory

Juha Oikkonen, university lecturer, infinitary logic, nonstandard analysis

Matti Pauna, Ph.D., using technology in mathematics education, non-wellfounded sets, set theory.

Jouko Väänänen, professor, set theory, model theory, dependence logic, foundations of mathematics.

Agatha Walczak-Typke, postdoctoral fellow, model theory, set theory, and algebra.

### Ph.D. students:

Teppo Kankaanpää, M.Sc.

Fan Yang, M.Sc.

Lauri Tuomi, M.Sc.

Kaisa Kangas, M.Sc.

Amir Oghbatalab, M.Sc.

Tapio Eerola, M.Sc.

Former members of the group can be found in the list of Ph.Ds from the Logic Group.

## Main Topics of Research

##### Members - Research - Publications - Links - Contact Info
MODEL THEORY
Research topics in model theory include
• homogeneous model theory i.e. model (stability) theory for classes of structures that consist of all elementary submodels of a large homogeneous structure.
• infinitary logic, especially transfinite Ehrenfeucht-Fraïssé games are studied and in connection with these games the group studies non-structure theory, i.e. develops methods for constructing complicated models. Problems in this topic are often problems on trees, which has led the group to study combinatorial properties of uncountable trees.
• abstract model theory, e.g. generalized quantifiers.
• set theoretic model theory, e.g. transfer principles and universality of regular reduced products.

SET THEORY
Topics in set theory include various aspects of trees and transfinite games, usually arising from model theoretic questions. A new interest is generalized cardinal invariants.

FINITE MODEL THEORY
In finite model theory the group has started a vigorous investigation of generalized quantifiers on finite models. New methods for establishing non-expressibility results have been developed with a keen eye on the possibility of obtaining new hierarchy results for finite models.

LOGIC AND ANALYSIS
In the topic of logic and analysis the group has studied
• first-order definability and expressibility in rings whose elements are complex analytic functions defined in a common domain, combining methods of classical complex analysis with those of logic and set theory.
• There is also some work done on Brownian motion with the help of nonstandard analysis.

OTHER TOPICS
Other topics include eLearning, dependence logic and philosophy of mathematics.

## A Selection of Publications

##### Members - Research - Publications - Links - Contact Info

Aapo Halko, Negligible subsets of the generalized Baire space , Ann. Acad. Sci. Fenn. Math. Dissertationes 107 (1996) 1-38.

Lauri Hella, Phokion G. Kolaitis, and Kerkko Luosto, Almost everywhere equivalence of logics in finite model theory, Bull. Symbolic Logic, 2(4):422-443, 1996.

Lauri Hella, Phokion G. Kolaitis, and Kerkko Luosto, How to define a linear order on finite models, Ann. Pure Appl. Logic, 87(3):241-267, 1997.

Lauri Hella, Kerkko Luosto, and Jouko Väänänen, The hierarchy theorem for generalized quantifiers, J. Symbolic Logic, 61(3):802-817, 1996.

Taneli Huuskonen, Tapani Hyttinen, and Mika Rautila, On the kappa-Cub Game on lambda and I[lambda], Arch. Math. Logic 1999, pp. 549-557.

Taneli Huuskonen, Tapani Hyttinen, and Mika Rautila, On potential isomorphism and non-structure, Arch. Math. Logic 43 (2004), pp. 85-120.

Taneli Huuskonen, Tapani Hyttinen, On definability of order in logic with choice, Proceedings, 16th Annual IEEE Symposium on Logic in Computer Science, Boston, MA, USA, pp. 167-172.

Taneli Huuskonen, Comparing notions of similarity for uncountable models, J. Symbolic Logic, 60(4):1153-1167, 1995.

Taneli Huuskonen, Observations about Scott and Karp trees, Ann. Pure Appl. Logic, 76(3):201-230, 1995.

Tapani Hyttinen and Heikki Tuuri, Constructing strongly equivalent nonisomorphic models for unstable theories, Annals of Pure and Applied Logic 52(1991) 203--248.

Tapani Hyttinen and Saharon Shelah, Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part B, The Journal of Symbolic Logic 60(1995) 1260--1272.

Tapani Hyttinen, Generalizing Morley's theorem, Mathematical Logic Quarterly 44(1998) 176--184.

Tapani Hyttinen and Saharon Shelah, Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic 103(2000) 201--228.

Tapani Hyttinen, Groups acting on geometries, Logic and algebra, Contemp. Math. 302(2002) 221--233.

Tapani Hyttinen and Saharon Shelah, Forcing a Boolean algebra with predesigned automorphism group, Proceedings of the American Mathematical Society, vol. 130, 2002, 2837-2844.

Tapani Hyttinen, Alex Hellsten, and Saharon Shelah, Potential isomorphism and semi-proper trees, Fundamenta Mathematicae, vol. 175, 2002, 127-142.

Tapani Hyttinen and Olivier Lessmann, A rank for the class of elementary submodels of a superstable homogeneous model, The Journal of Symbolic Logic, vol. 67, 2002, 1469-1482.

Tapani Hyttinen, Interpreting groups inside modular strongly minimal homogeneous models, Journal of Mathematical Logic, vol. 3, 2003, 127-142.

Tapani Hyttinen, Sy D. Friedman and Mika Rautila, Classification theory and 0#, The Journal of Symbolic Logic, vol. 68, 2003, 580-588.

Tapani Hyttinen and Matti Pauna, On non-wellfounded sets as fixed points of substitution, Notre Dame Journal of Formal Logic, vol. 42, 2003, 23-40.

Tapani Hyttinen, Finiteness of U-rank implies simplicity in homogeneous structures, Mathematical Logic Quarterly, vol. 49, 2003, 576-578.

Tapani Hyttinen, Finitely generated substructures of a homogeneous structure, Mathematical Logic Quarterly, vol. 50, 2004, 77-98.

Tapani Hyttinen, Taneli Huuskonen and Mika Rautila, Potential isomorphism and non-structure, Archive for Mathematical Logic, vol. 43, 2004, 85-120.

Tapani Hyttinen and Gabriel Sandu, Truth and definite truth, in: Z. Adamowitz et al. (ed.) Provinces of logic determined, Essays in the memory of Alfred Tarski, Annals of Pure and Applied Logic, vol. 126, 2004, 49-55.

Tapani Hyttinen, Types in abstract elementary classes, Notre Dame Journal of Formal Logic, vol. 45, 2004, 99-108.

Tapani Hyttinen and Gabriel Sandu, Deflatonism and arithmetical truth, Dialectica, vol. 58, 2004, 413-426.

Juliette Kennedy and Mark van Atten, Gödel's Modernism: On Set-Theoretic Incompleteness, Graduate Faculty Philosophy Journal, volume 25, number 2, 2004

Juliette Kennedy and Mark van Atten, On the Philosophical Development of Kurt Gödel, Bulletin of Symbolic Logic, Volume 9, Issue 4, December 2003. pages 425-476.

Juliette Kennedy, On embedding models of arithmetic into reduced powers, 8th Workshop on Logic, Language, Informations and Computation---WoLLIC'2001 (Brasília). Mat. Contemp. 24 (2003), 91--115.

Juliette Kennedy and Saharon Shelah, On embedding models of arithmetic of cardinality aleph_1 into reduced powers, Fundamenta Mathematicae 176(2003).

Juliette Kennedy and Saharon Shelah, On regular reduced products, J. Symbolic Logic 67(2002) 1169--1177.

Kerkko Luosto, Ultrametric spaces bi-Lipschitz embeddable in ${\bf R}\sp n$, Fund. Math., 150(1):25-42, 1996.

Kerkko Luosto, Ramsey theory is needed for solving definability problems of generalized quantifiers, Lecture Notes in Computer Science 1754 (1999), 121-134.

Juha Oikkonen, Undefinability of $\kappa$-well-orderings in $L\sb {\infty\kappa}$, J. Symbolic Logic, 62(3):999-1020, 1997.

J. Väänänen and D. Westerståhl, On the expressive power of monotone natural language quantifiers over finite sets, Journal of Philosophical Logic, 31(2002), 327--358.

B. Velickovic and J. Väänänen, Games played on partial isomorphisms, Archive for Mathematical Logic 43:1, 19-30, 2004.

J. Stavi and J. Väänänen, Reflection principles for the continuum. Logic and Algebra, ed. Yi Zhang, pp. 59-84, Contemporary Mathematics, Vol 302, AMS, 2002

S. Shelah and J. Väänänen, A Note on Extensions of Infinitary Logic, Archive for Mathematcial Logic, 44:1, 63-69, 2005.

T. Hyttinen, S. Shelah and J. Väänänen, More on the Ehrenfeucht-Fraisse game of length omega_1, Fundamenta Mathematicae, 175 (2002), no. 1, 79--96.

J. Väänänen, Pseudo-finite model theory, Matematica Contemporanea vol 24, 2003, 169-183.

R. Parikh and J. Väänänen, Finite information logic, Annals of Pure and Applied Logic vol. 134, 2005, 83-93

J. Väänänen, Barwise, Abstract model theory and generalized quantifiers, Bulletin of Symbolic Logic, vol. 10:1, 2004, 37-53.

M. Dzamonja and J. Väänänen, A family of trees with no uncountable branches, Topology Proceedings vol. 28:1, 2004, 113-132.

Jouko Väänänen, On the semantics of informational independence, Logic Journal of the Interest Group in Pure and Applied Logics, 10:3, 337-350, 2002.

Jouko Väänänen, Second order logic and foundations of mathematics, Bulletin of Symbolic Logic, Volume 7, Issue 4, December, 2001.

Saharon Shelah and Jouko Väänänen, Stationary sets and infinitary logic, Journal of Symbolic Logic, 65:1311-1320, 2000.

Stevo Todorcevic and Jouko Väänänen, Trees and Ehrenfeucht-Fraisse games, Annals of Pure and Applied Logic, 100, 69-97, 1999.

Jouko Väänänen, Unary quantifiers on finite models, J. Logic Lang. Inform., 6(3):275-304, 1997.

  Department of Mathematics       telephones (358-9)-191 51500