Research program: Large Cardinals and Strong Logics

The research program on Large Cardinals and Strong Logics will take place at the Centre de Recerca Matemàtica, September 5–December 16, 2016.

Many natural mathematical concepts cannot be expressed in first-order logic but need stronger logics. Among such concepts are the freeness of a group, separability of a space, completeness of an order, etc. This led to the introduction of the concept of a generalized quantifier, which made it possible to compare model-theoretic and set-theoretic definability of various mathematical concepts. It turned out that there is a close connection between the two.

By a strong logic we mean model-theoretically defined extensions of first-order logic, such as first-order logic with generalized quantifiers, infinitary logics, second-order logic, as well as higher-order logics. The study of strong logics runs immediately into questions that depend essentially on set-theoretical assumptions beyond the standard ZFC axioms, such as infinitary combinatorial principles and the existence of large cardinals. It is therefore crucial to be able to pinpoint the position of a given strong logic in the set-theoretical definability hierarchy, thus helping us understand better the set-theoretical nature of the logic, and therefore of the mathematical notions it can express.
Objectives

This program will bring to the CRM​ a diverse group of international high-level researchers working in strong logics, large cardinals, the foundations of set theory, and the applications of set-theoretical methods in other areas of mathematics, such as algebra, set-theoretical topology, category theory, algebraic topology, homotopy theory, C*-algebras, measure theory, etc. In all these areas there are not only direct set-theoretical applications but also new results and methods, which are amenable to the expressive power of strong logics.

The Scientific Committee includes: J. Bagaria, M. Magidor, and J. Väänänen.

For further information, visit the webpage http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/IRP-Large-Cardinals-and-Strong-Logics.aspx

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