Department of Foundations of Mathematics

Head:

• Zofia Adamowicz (Professor)
  email
  pok. 506, tel.: 22 5228 172

Staff:

• Ryszard Frankiewicz (Professor)
  email
  Division in Wrocław
• Piotr Koszmider (Associate Professor)
  email
  pok. 505, tel.: 022 5228 148
• Maciej Malicki (Assistant Professor)
• Marcin Sabok (Assistant Professor)
  email
  pok.420, tel.: 22 5228 167

Phd Students:

• Mikołaj Krupski
  email
• Joanna Ochremiak
  email
• Sławomir Szczepaniak
  email

About the Department

Research of the Section involves a rather wide spectrum of matters which are connected with the foundations of mathematics, such as set theory, elements of functional analysis, elements of real analysis, foundations of arithmetic, cathegorical logic.

Zofia Adamowicz

Zofia Adamowicz is working in foundations of arithmetic. Her main research is in "bounded arithmetic" together with its links to computational complexity and the P=NP problem. Recent research concerns the power of the exponential function in arithmetic, e.g. its influence on the existence of end extensions of models.

Some recent papers:

• A contribution to the end-extension problem and the Π₁ conservativeness problem, Ann. Pure Appl. Logic 61, pp. 3-48, 1993.
• Existentially Closed Structures and Gödel's Second Incompleteness Theorem (with T. Bigorajska), J. Symb. Log, 66(1), pp. 349-356, 2001.
• On Herbrand consistency in weak arithmetic (with P. Zbierski), Arch. Math. Logic, 40(6), pp. 399-413, 2001
• Herbrand Consistency and Bounded Arithmetic, Fundamenta Mathematica, 171, pp. 279-292, 2002
• On complexity reduction of Σ₁ formulas (with P. Zbierski), Arch. Math. Logic 42(1), pp. 45-58, 2003.
• Well-behaved principles alternative to bounded induction (with L. A. Kołodziejczyk), Theor. Comput. Sci. 322(1), pp. 5-16, 2004.
• Partial collapses of the Sigma₁ complexity hierarchy in models for fragments of bounded arithmetic (with L. A. Kołodziejczyk and P. Zbierski), Ann. Pure Appl. Logic, 145(1), pp. 91-95, 2007.
• Lower bounds for the unprovability of Herbrand consistency in weak arithmetics (with K. Zdanowski), submitted.

Also she is a co-author of a handbook of logic:

• Logic of Mathematics (with P. Zbierski), Wiley, 1997.

Ryszard Frankiewicz

His research interest includes: infinite combinatorics, applications of set theory to mathematical analysis, in particular: the algebra P(ω)/fin, the groups of automorphisms of Boolean algebras, unconditional bases in Banach spaces, discontinuous homomorphisms of Banach algebras.

Some recent publications:

• Hausdorff Gaps and Limits, Stud. Logic Found. Math. 132, North-Holland, 1994 (with P. Zbierski).
• Borel liftings of the measure algebra and the failure of the continuum hypothesis, Proc. Amer. Math. Soc. 120 (1994), 1247-1250 (with T. Carlson and P. Zbierski).
• Nonaccessible filters in measure algebras and functionals on L_∞(Λ)^*, Studia Math. 108 (1994), 191-200 (with G. Plebanek).
• On asymptotic density and uniformly distributed sequences, Studia Math. 119 (1996), 17-26 (with G. Plebanek).
• Convex combinations and weak^* null sequences, Bull. Polish Acad. Sci. 45 (1997), 221-225 (with G. Plebanek).
• On closed P-sets without ccc in the space ω^*, Israel J. Math. (with S. Shelah and P. Zbierski), to appear.
• Fat P-sets in the space ω^*, J. Symbolic Logic (with P. Zbierski), to appear.

Piotr Koszmider

His research is focused on developing and applications of the methods of combinatorial set theory and logic such as forcing, stepping up, anti-Ramsey results, bookkeeping principles in analysis and topology in particular in Banach spaces, weak and weak* topology and in algebras of operators.

Some recent publications:

• A. Aviles, P. Koszmider, A continuous image of a Radon-Nikodym compact space which is not Radon-Nikodym; To appear in Duke Math. J.
• A. Avilés, P. Koszmider; A Banach space in which every injective operator is surjective. To apepar in Bull. London Math. Soc.
• Piotr Koszmider; On large indecomposable Banach spaces; J. Funct. Anal. 264 (2013), no. 8, 1779-805
• Jesus Ferrer, Piotr Koszmider, Wieslaw Kubis; Almost disjoint families of countable sets and separable complementation properties; J. Math. Anal. Appl. 401 (2013), no. 2, 939-949
• Christina Brech, Piotr Koszmider; On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts; Proc. Amer. Math. Soc. 141 (2013), 1267-1280
• Piotr Koszmider; A C(K) Banach space which does not have the Schroeder-Bernstein property; Studia Math. 212 (2012), 95-117
• Christina Brech, Piotr Koszmider; On universal Banach spaces of density continuum, Israel J. Math. 190 (2012), 93-110

Czesław Ryll-Nardzewski

His research interests include mathematical logic, set theory, descriptive set theory, topology, probability theory and stochastic processes, functional and harmonic analysis, real analysis, ergodic theory, differential equations. He is an author (or co-author) of many well known and important results, including, to mention but a few: Ryll-Nardzewski theorem on categoricity, the theorem on impossibility of finite axiomatization of arithmetic, Ryll-Nardzewski fixed point theorem, Kuratowski and Ryll-Nardzewski selector theorem.

Konrad Zdanowski

His research interests includes arithmetics with bounded induction, finite model theory, intuitionistic logic, philosophy of mathematics.

Publications:

• On Spectra of formulae with Henkin Quantifiers (with J. Golińska), in Philosophical Dimensions of Logic and Science, Proc.of LMPhSc 1999, ed. A. Rojszczak, J. Cachro, G. Kurczewski, Kluwer, 2003, pp. 29--45.
• Degrees of logics with Henkin Quantifiers (with M. Mostowski), in Archive for Mathematical Logic,  43(2004), pp.691--702.
• Theories of arithmetics in finite models (with M. Krynicki), in Journal of Symbolic Logic, 70(2005), pp. 1--28.
• FM-representability and beyond (with M. Mostowski), Proc. of  CiE 2005, Lecture Notes in Computer Science vol. 3526, Springer, pp.358--367.
• Coprimality in finite models (with M. Mostowski), Proc. of CSL 2005, Lecture Notes in Computer Science, vol. 3634, Springer,      pp.  263--275.
• Finite Arithmetics (with M. Krynicki and M. Mostowski), Fundamenta Informaticae, vol. 81(2007), pp. 183--202.
• The Intended Model of Arithmetic. An Argument from Tennenbaum's Theorem (with P. Quinon), in Computation and Logic in the Real World, CiE 2007, Local Proceedings, ed. S.B. Cooper, B. Loewe i A. Sorbi, 2007.
• Undecidability and concatenation (with A. Grzegorczyk), in Andrzej Mostowski and Foundational Studies, ed. W. Marek,                A. Ehrenfeucht, M. Srebrny, IOS Press, Amsterdam, 2008.
• On the  second order intuitionistic propositional logic without a universal quantifier, submitted.
• Lower bounds for the unprovability of Herbrand consistency in weak arithmetics (with Z. Adamowicz), submitted.
• On a question of Andreas Weiermann (with H. Kotlarski), submitted.

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