

1978  Diploma, University of Bonn  1979  Master of Arts, University of California, Berkeley, CA, USA  1984  PhD, University of Freiburg  1984  1987  FeodorLynenFellowship, Alexander von Humboldt Foundation, and   Junior Research Fellow, Wolfson College, Oxford, England, UK  1987  1990  Assistant Professor (C1), Habilitation, University of Freiburg  Since 1990  Professor (C3), University of Bonn 


My set theoretical research focusses around the construction and analysis of models of set theory with various combinatorial properties, using the methods of forcing, inner models, and symmetric models. My main interest is on models having strong closure properties in the form of large cardinals like measurable cardinals and canonical strengthenings. Model constructions allow to classify set theoretic properties in terms of the existence of large cardinals: A model with large cardinals is extended by forcing to a model of the combinatorial property; conversely assuming the property one defines an inner model of set theory with large cardinals.
Recent results of this type concern “small” measurable cardinals within the bounded Gitik model (with A. Apter and I. Dimitriou), model theoretic properties about the existence of elementary substructures with
cardinality constraints (with A. Apter and I. Dimitriou) or the existence of forcing extensions in which successor cardinals of the ground model become singular (with D. Adolf and A. Apter). Sometimes large cardinals can be eliminated: we constructed 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).
The model of constructible sets by K. Gödel can be obtained in several ways. I developed the approach by 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 BlumShubSmale machines.
The foundations of mathematics encompass the (natural) language of mathematics. The Naproche system developed in the logic group shows that natural language processing can be applied to mathematics: Naproche prototypically accepts proof texts in natural mathematical language and checks their correctness (with M. Cramer). For longer texts though we are experiencing a combinatorial explosion in the current setup since the background automatic theorem prover is given too many premises for its proof search.
In future research the settheoretical models mentioned above will be analyzed further. What remains of the ground model large cardinal properties in the Gitik model? Are the strongly compact cardinals of the ground model still Rowbottom cardinals? What is the cardinal arithmetic of infinite sums and products in the model with 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? Work on the minimality of Prikry forcing with Gitik and Kanovei which shows that all nontrivial subforcings of Prikry forcing are themselves Prikry forcings is to be finalized.
I shall use ordinal computability for the fine structural analysis of constructible sets. One can use ordinal computability theory to reconstruct an existing but cumbersome fine structure theory of J. Silver. There should however be more a direct approaches in which the typical objects of fine structure can be obtained by computations. Constructible models are also relevant in the project to generalized descriptive set theory. Such models provide wellordering of low definitional complexity, and hence counterexamples to regularity properties of low complexity.
The Naproche approach shall be extended to natural mathematical argumentation in collaborations with the Isabelle community (L. Paulson and M. Wenzel) and A. Paskevich (SAD system). We shall pursue the thesis that the combination of Naproche techniques with largescale technical systems like Isabelle may overcome the complexity problems indicated above by intelligent premise selection. We shall apply the methods to logical and settheoretical texts. Results of these experiments can yield philosophical insights into the nature of mathematical proofs.


DFG project “Complexity and Definability at Higher Cardinals”


Former Research Area L Algorithms in transfinite set theory:
Ordinal computability provides a unifying spectrum of computabilities, parameterized by ordinal time and space bounds, where the computable sets correspond to Borel, , Gödelconstructible sets and other classes. We prove fundamental properties of these classes via computability. A fine structure theory for the constructible universe can be defined using ordinal algorithms. Other processes like dynamic systems or BlumShubSmale computations will be continued into the infinite ordinals.
Formal mathematics:
In the Naproche project (Natural language proof checking), we connect ordinary mathematical texts with fully formal mathematics. The efficient use of automatic theorem provers for proof checking depends on the right choice of proof obligation sent to the prover. We study selection and preprocessing algorithms based on heuristics and natural language triggers. We are reformulating Landau's Grundlagen der Analysis into human readable and computer checked formats.
We shall combine Naproche techniques with established powerful formal mathematics systems.
Infinitary combinatorics:
We shall examine cardinal arithmetic for singular cardinals without assuming the axiom of choice, expanding on joint work with Apter and Gitik. We conjecture that infinitary cardinal exponentiation can take arbitrary cardinal values as long as some basic monotonicity is respected. This contrasts with the singular cardinal behaviour if the axiom of choice is assumed. 


[ 1] 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): 225231 2016 DOI: 10.1002/malq.201400050[ 2] 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): 471486 2014 DOI: 10.1002/malq.201110007[ 3] Moti Gitik, Peter Koepke
Violating the singular cardinals hypothesis without large cardinals Israel J. Math. , 191: (2): 901922 2012 DOI: 10.1007/s118560120028x[ 4] Peter Koepke, Julian J. Schlöder
The Gödel completeness theorem for uncountable languages Formalized Mathematics , 20: : 199203 2012[ 5] P. Koepke, P. D. Welch
Global square and mutual stationarity at the {\aleph_n} Ann. Pure Appl. Logic , 162: (10): 787806 2011 DOI: 10.1016/j.apal.2011.03.003[ 6] Peter Koepke
Turing computations on ordinals Bulletin of Symbolic Logic , 11: (3): 377397 2005[ 7] Peter Koepke
Extenders, embedding normal forms, and the MartinSteeltheorem J. Symbolic Logic , 63: (3): 11371176 1998 DOI: 10.2307/2586731[ 8] Sy D. Friedman, Peter Koepke
An elementary approach to the fine structure of L Bull. Symbolic Logic , 3: (4): 453468 1997 DOI: 10.2307/421099



2009  Computability in Europe, Heidelberg  2009  Effective Mathematics of the Uncountable 2009, CUNY, New York, USA  2010  Set Theory, Classical and Constructive, Amsterdam, Netherlands  2010  Set Theory, Model Theory, Generalized Quantifiers and Foundations of Mathematics, Helsinki, Finland  2011  14th Congress of Logic, Methodology and Philosophy of Science, Nancy, France  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 


Heike Mildenberger (1998), now Professor, University of Freiburg
Benedikt Löwe (2005), now Professor, University of Amsterdam, Netherlands, and University of Hamburg


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, Mathematics, University of Konstanz
Marcos Cramer (2013): “Proofchecking mathematical texts in controlled natural language”,
now Research Assistant, Computer Science, University of Luxembourg
Benjamin Seyfferth (2013): “Three models of ordinal computability”,
now Coordinator of Studies, Mathematics, University of Darmstadt


 Master theses: 1
 Diplom theses: 55, currently 5
 PhD theses: 12, currently 5


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