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Mittwoch, 7.2.18, 16:30-17:30, Raum 404, Eckerstr. 1
Compactness and reflection in mathematics
Mittwoch, 7.2.18, 16:30-17:30, Raum 404, Eckerstr. 1
Abstract: One of the most fruitful research area in set theory is the study of the so-called reflections principles'. Roughly speaking, by reflection principle we mean a combinatorial statement of the following form: given a structure S (e.g. a stationary set, a tree, a graph, a groups ...) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. Compactness is dual to reflection, namely bycompactness property' we mean roughly a statement of the following form: given a structure S and a property P in the language of the structure, if every smaller substructure has the property P, then S satisfies P as well. \n\nMany interesting mathematical problems can be formulated as compactness problems; for instance, there is an extensive literature on the compactness problem for the property of being a free group: given a group G, suppose that every small subgroup (i.e. of smaller size) is free, is G itself free? This problem is independent from ZFC and the answer depends on the cardinality of the group. \n\nStrong forms of reflection are typically associated with large cardinals axioms, which therefore imply interesting compactness results. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut. \n\n
Compactness and reflection
Mittwoch, 7.2.18, 17:30-18:30, Raum 404, Eckerstr. 1
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