Mathematical Sciences: Inevitability in Model Theory

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9403701 Laskowski Dr. Michael Laskowski will continue his research in the field of model theory, a branch of mathematical logic. He expects to obtain more information as to the possible mechanisms making a type inevitable in the setting of superstable theories that do not have the Dimensional Order Property (DOP). Laskowski expects to complete his classification of those countable theories for which there are two non-isomorphic models that can be forced isomorphic by a forcing with the countable chain condition. Finally, he will continue his research into an inherent asymmetry between two new methods of constructing models of small, superstable theories. Most of the field of model theory (as a branch of mathematical logic) is concerned with developing methods for building algebraic structures with certain properties. The problem of whether there is a structure having an element satisfying an infinite set of properties (i.e., realizing a type) is well understood: such a structure exists if and only if, for every finite subset of the type, there is an element satisfying each of these properties. However, it is a much more subtle question to ask whether a given type is inevitable, i.e., whether EVERY structure contains an element realizing the type. Without adding additional restrictions to the class of structures under consideration, it is known that there can be no satisfactory criterion for which types are inevitable. Laskowski will continue his research on what additional constraints must be placed on the algebraic structure to yield a natural characterization of inevitability. ***