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Set theory seminarThursday, July 7, 20165:00 pmGC 4214.03Math Thesis Room

Laura Fontanella

Reflection of stationary sets, tree property and square

Hebrew University of Jerusalem

One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, 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. 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.