Department of Foundations of Mathematics
- Michal Doucha (Assistant Professor)
- Ryszard Frankiewicz (Professor)
Division in Wrocław
- Tomasz Kania (Assistant Professor)
pok. 419, tel.: 22 522 82 38
- Piotr Koszmider (Professor)
pok. 505, tel.: 22 5228 148
- Marcin Sabok (Assistant Professor)
pok.420, tel.: 22 5228 167
About the Department
Research of the Section involves a rather wide spectrum of matters which
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 is working in foundations of arithmetic.
Her main research
"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.
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),
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
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.
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
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
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
fixed point theorem,
Kuratowski and Ryll-Nardzewski
His research interests includes arithmetics with bounded induction,
finite model theory, intuitionistic logic, philosophy of mathematics.
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.
- 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,
- 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.