Past Logic Seminar

7 March 2008
14:15
Hans Adler
Abstract
I will explain the connection between Shelah's recent notion of strongly dependent theories and finite weight in simple theories. The connecting notion of a strong theory is new, but implicit in Shelah's book. It is related to absence of the tree property of the second kind in a similar way as supersimplicity is related to simplicity and strong dependence to NIP.
29 February 2008
14:15
Alexey Muranov
Abstract
Bardakov and Tolstykh have recently shown that Richard Thompson's group $F$ interprets the Arithmetic $(\mathbb Z,+,\times)$ with parameters. We consider a class of infinite groups of piecewise affine permutations of an interval which contains all the three groups of Thompson and some classical families of finitely presented infinite simple groups. We have interpreted the Arithmetic in all the groups of this class. In particular we have obtained that the elementary theories of all these groups are undecidable. Additionally, we have interpreted the Arithmetic in $F$ and some of its generalizations without parameters. This is a joint work with Tuna Altınel.
15 February 2008
14:15
Ayhan Gunaydin
Abstract
There is a well-behaving class of dense ordered abelian groups called "regularly dense ordered abelian groups". This first order property of ordered abelian groups is introduced by Robinson and Zakon as a generalization of being an archimedean ordered group. Every dense subgroup of the additive group of reals is regularly dense. In this talk we consider subgroups of the multiplicative group, S, of all complex numbers of modulus 1. Such groups are not ordered, however they have an "orientation" on them: this is a certain ternary relation on them that is invariant under multiplication. We have a natural correspondence between oriented abelian groups, on one side, and ordered abelian groups satisfying a cofinality condition with respect to a distinguished positive element 1, on the other side. This correspondence preserves model-theoretic relations like elementary equivalence. Then we shall introduce a first-order notion of "regularly dense" oriented abelian group; all infinite subgroups of S are regularly dense in their induced orientation. Finally we shall consider the model theoretic structure (R,Gamma), where R is the field of real numbers, and Gamma is dense subgroup of S satisfying the Mann property, interpreted as a subset of R^2. We shall determine the elementary theory of this structure.
18 January 2008
14:15
Itai Ben Yaacov
Abstract
H. Jerome Keisler suggested to associate to each classical structure M a family of "random" structures consisting of random variables with values in M . Viewing the random structures as structures in continuous logic one is able to prove preservation results of various "good" model theoretic properties e.g., stability and dependence, from the original structure to its randomisation. On the other hand, simplicity is not preserved by this construction. The work discussed is mostly due to H. Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).

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