Profile
Profile

Prof. Dr. Peter Koepke

E-mail: koepke(at)math.uni-bonn.de
Phone: +49 228 73 2206
Fax: +49 228 73 62249
Homepage: http://www.math.uni-bonn.de/people/koepke/
Room: 4.005
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Former Research Area L (Leader)
Research Area KL
Date of birth: 31.May 1954

Academic Career

1978

Diploma, University of Bonn

1979

Master of Arts, University of California, Berkeley, CA, USA

1984

PhD, University of Freiburg

1984 - 1987

Feodor-Lynen-Fellowship, Alexander von Humboldt Foundation, and

Junior Research Fellow, Wolfson College, Oxford, England, UK

1987 - 1990

Assistant Professor (C1), Habilitation, University of Freiburg

2014

Visiting Fellow and Life Member, Clare Hall, Cambridge, UK

Since 1990

Professor (C3), University of Bonn

Research Profile

My set theoretical research focusses around the construction and analysis of models of set theory with various combinatorial properties, using methods of forcing, inner models, and symmetric models. My main interest is on models having strong closure properties expressed by the existence of large cardinals like measurable and stronger cardinals. Model constructions allow to classify set theoretic properties in terms of large cardinals: A model with large cardinals is extended by forcing to a model of the combinatorial property; conversely assuming such a property one defines inner models of set theory with large cardinals.

The following questions are representative of my current research projects in axiomatic set theory: What remains of the ground model large cardinal properties in M. Gitik‘s model in which every cofinality is countable? What is the cardinal arithmetic of infinite sums and products in a model that I constructed with A. Fernengel ? How does Shelah's theory of possible cofinalities behave in that model? Can the model be modified so that the axiom of choice holds for countable families? I shall finalize work on the minimality of Prikry forcing with Gitik and Kanovei. The method of ordinal computability which I have developed will be employed in the fine structural analysis of Gödel‘s model of constructible sets.

In formal mathematics I shall further develop A. Paskevich‘s SAD system which is orientated towards natural mathematical language and argumentation. Based on previous experience with the Naproche system we are adding state-of-the-art natural language processing to SAD.

Research Projects and Activities

DFG project “Complexity and Definability at Higher Cardinals”
2015 - 2017

Contribution to Research Areas


Research Area KL
My main results in axiomatic set theory, with co-authors, deal with “small” measurable cardinals within the bounded Gitik model [1], model theoretic properties about the existence of elementary substructures with cardinality constraints [2] or forcing extensions in which measurable cardinals or successor cardinals of the ground model become singular ([3], [4]). Sometimes large cardinals can be eliminated: we construct models of set theory without the axiom of choice in which the generalized continuum hypothesis formulated by F. Hausdorff can be violated in rather arbitrary ways (with A. Fernengel).

I developed the theory of ordinal computability, combining Turing computability and uncountable set theory. Calibrating certain parameters of ordinal computability one obtains initial segments of Gödel's model of various heights. With A. Morozov I determined the segment corresponding to infinite time Blum-Shub-Smale machines [5].
In formal mathematics, we improved an earlier formalization of Gödel‘s completeness theorem to arbitrary languages [6]. I participated in philosophical discussions on the future impact of computer-supported formal mathematics.

Selected Publications

[1] Arthur W. Apter, Ioanna M. Dimitriou, Peter Koepke
The first measurable cardinal can be the first uncountable regular cardinal at any successor height
MLQ Math. Log. Q. , 60: (6): 471--486
2014
[2] Arthur W. Apter, Ioanna M. Dimitriou, Peter Koepke
All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters
MLQ Math. Log. Q. , 62: (3): 225--231
2016
[3] Peter Koepke, Karen Räsch, Philipp Schlicht
A minimal Prikry-type forcing for singularizing a measurable cardinal
J. Symbolic Logic , 78: (1): 85--100
2013
[6] Peter Koepke, Julian J. Schlöder
The Gödel completeness theorem for uncountable languages
Formalized Mathematics , 20: : 199--203
2012
[7] Moti Gitik, Peter Koepke
Violating the singular cardinals hypothesis without large cardinals
Israel J. Math. , 191: (2): 901--922
2012
[8] P. Koepke, P. D. Welch
Global square and mutual stationarity at the {\aleph_n}
Ann. Pure Appl. Logic , 162: (10): 787--806
2011
[9] Peter Koepke
Turing computations on ordinals
Bulletin of Symbolic Logic , 11: (3): 377--397
2005
[10] Peter Koepke
Extenders, embedding normal forms, and the Martin-Steel-theorem
J. Symbolic Logic , 63: (3): 1137--1176
1998
[11] Sy D. Friedman, Peter Koepke
An elementary approach to the fine structure of L
Bull. Symbolic Logic , 3: (4): 453--468
1997

Publication List

MathSciNet Publication List (external link)

Selected Invited Lectures

2013

CUNY Logic Workshop, New York, USA

2013

Proof 2013, Bern, Switzerland

2013

Mal'cev Meeting, Novosibirsk, Russia

2014

60th birthday conference of Philip Welch, Bristol, England, UK

2015

Set Theory, Carnegie Mellon University, Pittsburgh, PA, USA

2015

Philosophy of Mathematics Seminar, Oxford, England, UK

2015

European Set Theory Conference, Cambridge, England, UK

2016

Menachem Magidor 70th Birthday Conference, The Hebrew University of Jerusalem, Israel

Habilitations

Heike Mildenberger (1998), now Professor, University of Freiburg

Benedikt Löwe (2005), now Professor, University of Amsterdam, Netherlands, and University of Hamburg

Selected PhD students

Ralf Schindler (1996): “The Core Model up to one Strong Cardinal”,
now Professor (C4), Mathematics, University of Münster

Merlin Carl (2011): “Alternative finestructural and computational approaches to constructibility”,
now Assistant Professor and Privatdozent, Mathematics, University of Konstanz

Benjamin Seyfferth (2013): “Three models of ordinal computability”,
now Coordinator of Studies, Mathematics, University of Darmstadt

Regula Krapf (2017): “Class forcing and second-order arithmetic”,
now Assistant Professor, Mathematics, University Koblenz-Landau

Supervised Theses

  • Master theses: 9, currently 5
  • Diplom theses: 60
  • PhD theses: 11, currently 2
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