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University of Oxford·Joan Bagaria
1:08:47
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Recent Results on Structural Reflection Principles
A presentation on recent advancements in Structural Reflection principles, exploring their variants and equivalences to various large cardinal axioms weaker than Vopenka's Principle.
Abstract. The general Structural Reflection (SR) principle asserts that for every definable, in the first-order language of set theory, possibly with parameters, class C of relational structures of the same type there exists an ordinal alpha that reflects C, i.e., for every A in C there exists B in C intersect V_alpha and an elementary embedding from B into A. In this form, SR is equivalent to Vopenka’s Principle (VP). In my talk I will present some different natural variants of SR which are equivalent to the existence some well-known large cardinals weaker than VP. I will also consider some forms of SR, reminiscent of Chang’s Conjecture, which imply the existence of large cardinal principles stronger than VP, at the level of rank-into-rank embeddings and beyond. The latter is a joint work with Philipp Lücke.