Zakład Podstaw Matematyki

Kierownik:

Pracownicy:

  • prof. dr hab. Ryszard Frankiewicz (profesor)
    email
  • dr hab. Adam Obtułowicz (docent)
    email
    pok. 424, tel.: 022 5228 235
  • prof. dr hab. Czesław Ryll-Nardzewski (profesor)
    email
    tel.: 071 320 21 10
  • dr Konrad Zdanowski (adiunkt)
    email
    610a, tel.: 022 5228 224

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.

Adam Obtułowicz

His research is in cathegorical logic.

Main publications:

  • Algebra of constructions I. The word problem for partial algebras, Inform. and Comput. 73 (1987), 129-173.
  • An algebraic approach to Martin-Loef type theory and the calculus of constructions, Math. Structures in Computer Sci. 3 (1993), 63-92.

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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