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Mini-workshop: Logic, Combinatorics and Independence Results
Organized by: Andrey I. Bovykin (1), Lorenzo Carlucci (2) and Andreas Weiermann (3)(1) Department of Computer Science, University of Liverpool, Chadwick Building, Peach Street, L69 3BX, LIVERPOOL, UNITED KINGDOM
(2) Dipartimento di Scienze Matematice e Informatiche, Universita di Siena, Pian dei Mantellini, 44, I-53100, SIENA, ITALY
(3) Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, P.O. Box 80010, 3508 TA, UTRECHT, NETHERLANDS
The mini-workshop ``Logic, Combinatorics and Independence results" took place on November 25 -- December 2, 2006.
The workshop was organized by Andrey Bovykin, Lorenzo Carlucci and Andreas Weiermann and attended by 16 participants:
\item Lev Beklemishev (Moscow)
\item Andrey Bovykin (Liverpool)
\item Wilfried Buchholz (Munchen)
\item Lorenzo Carlucci (Rome)
\item Lev Gordeev (T\"ubingen)
\item Henryk Kotlarski (Warszawa)
\item Alberto Marcone (Udine)
\item Joseph Mileti (Chicago)
\item Antonio Montalb\'an (Wellington)
\item Eran Omri (Be'er-Sheva)
\item Michael Rathjen (Leeds)
\item Sergei Tupailo (Tallinn)
\item Stanley Wainer (Leeds)
\item Andreas Weiermann (Gent)
\item Alan Woods (Crawley)
\item Konrad Zdanowski (Warszawa)
There were 16 one-hour talks, two problem sessions and many one-to-one and small group discussions.
The workshop brought together researchers specialising in several connected disciplines:
Reverse Mathematics (Alberto Marcone, Joseph Mileti, Antonio Montalb\'an),
Proof Theory (Lev Beklemishev, Wilfried Buchholz, Lev Gordeev, Michael Rathjen, Sergei Tupailo, Stanley Wainer, Andreas Weiermann),
WQO theory (Alberto Marcone, Antonio Montalb\'an, Lev Gordeev, Andreas Weiermann),
Models of Arithmetic (Andrey Bovykin, Henryk Kotlarski, Konrad Zdanowski, Alan Woods),
weak arithmetics (Lev Beklemishev, Alan Woods, Konrad Zdanowski),
logical aspects of finite Ramsey Theory (Andrey Bovykin, Lorenzo Carlucci, Henryk Kotlarski, Joseph Mileti, Eran Omri, Andreas Weiermann).
However, the central theme of the workshop was first-order unprovable statements and statements of large logical strength.
The subject originated in the late 1970s in the work of several mathematicians,
most notably Jeff Paris and Harvey Friedman and attracted a large community of researchers at that time.
The discoveries of the Paris-Harrington Principle and unprovability of Kruskal's Theorem
provided, fourty years after G\"odel's theorems,
the first examples of mathematically natural unprovable statements.
Since then, many other examples were found in Ramsey Theory,
Graph Theory, well-quasi-order theory and other subjects.
One of the main objectives
of the workshop was to revive research in this area, especially in
view of some spectacular recent developments.
These developments revealed deep connections between
the study of logical strength and several mathematical disciplines:
Analytic Combinatorics, Graph Theory, Tauberian Theory, Number Theory, Dynamical Systems.
Another objective of the workshop was to stimulate communication and
joint research between researchers from different sub-areas of the subject
(Ordinal Analysis, Reverse Mathematics, Models of Arithmetic).
All these different areas were represented at the workshop
by leading researchers. The workshop was very successful in setting grounds
for comparison and interaction of methods from these areas.
The two open problem sessions resulted in a list of problems of
common interest and in better understanding of possible directions for future research.
The talks varied from reports on recent results and proposals of new
general approaches to discussions of new strategies to tackle
long-standing open problems. Time allowed for free informal discussion
and research in small groups that will eventually result in publications.
We would like to thank the Oberwolfach Institute for the wonderful opportunity to hold a meeting
there and for providing NSF travel grants to some participants. We also thank all the staff working in
the institute for the pleasant experience we all had during our week in Oberwolfach Institute.
We would like to thank the Oberwolfach Institute for the wonderful opportunity to hold a meeting there and for providing NSF travel grants to some participants. We also thank all the staff working in the institute for the pleasant experience we all had during our week in Oberwolfach Institute.
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