Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments Program and Practical Information Contents

In 1990, W. Hugh Woodin introduced Omega-logic as an approach to truth in the universe of all sets inspired by recent work on large cardinals and determinacy. In Omega-logic statements are valid if, roughly, they hold in every forcing extension of the universe, or its sufficiently rich initial segments. Thus, Omega-logic is the logic of generic absoluteness. In this short course we will start with a brief introduction to forcing, large cardinals, and generic absoluteness. We will then describe the basic features of Omegalogic, leading to the formulation of the Omega-conjecture and a discussion of its relevance in current set-theoretic research. Suggested Background Reading: [BCL06], [Jec03], [Kan03], [Kun80], [Woo99], [Woo01]. 2.2 Fernando Ferreira (Lisbon) An Overview of Predicativity Abstract: The classical discussion on predicativity: Poincaré and Russell. The logicist project. Ramification and reducibility. The consistency of Frege’s predicative system. The impredicativity of induction. What can be done in a strict predicative system? Finite reducibility. Predicativity given the natural numbers. Weyl’s approach. Kreisel’s proposals. The hyperarithmetic sets. Predicative provability and the Feferman-Schütte proof-theoretical analysis. Predicative mathematics. A working hypothesis concerning the indispensability arguments. The non-realistic cast of predicativism. A predicative logic? Reverse mathematics and the limits of predicativity. Impredicativity and realism. Suggested Background Readings: [Fef05], [Hec96] (on a first reading, perhaps skip section 3), [Par02], [Sim02]. The classical discussion on predicativity: Poincaré and Russell. The logicist project. Ramification and reducibility. The consistency of Frege’s predicative system. The impredicativity of induction. What can be done in a strict predicative system? Finite reducibility. Predicativity given the natural numbers. Weyl’s approach. Kreisel’s proposals. The hyperarithmetic sets. Predicative provability and the Feferman-Schütte proof-theoretical analysis. Predicative mathematics. A working hypothesis concerning the indispensability arguments. The non-realistic cast of predicativism. A predicative logic? Reverse mathematics and the limits of predicativity. Impredicativity and realism. Suggested Background Readings: [Fef05], [Hec96] (on a first reading, perhaps skip section 3), [Par02], [Sim02]. 2.3 Luca Incurvati (Cambridge) and Hannes Leitgeb (Munich), Groundedness in Set Theory and Semantics Abstract: This three-lectures course will focus on groundedness in set theory and semantics. Topics to be discussed on the set-theoretic side will be: the iterative conception of sets; differences and connections between various groundedness assumptions for sets; the issue of the justification of these assumptions and of assumptions violating groundedness; the connection, if any, between groundedness and the logic of set theory. And on the semantic side: grounded instances of the Tarskian T-scheme; semantic dependency; dependency vs. membership; and a note on grounded abstraction. Suggested Background Readings: [Inc11], [Lei05], [Lin08], [Rie00]. This three-lectures course will focus on groundedness in set theory and semantics. Topics to be discussed on the set-theoretic side will be: the iterative conception of sets; differences and connections between various groundedness assumptions for sets; the issue of the justification of these assumptions and of assumptions violating groundedness; the connection, if any, between groundedness and the logic of set theory. And on the semantic side: grounded instances of the Tarskian T-scheme; semantic dependency; dependency vs. membership; and a note on grounded abstraction. Suggested Background Readings: [Inc11], [Lei05], [Lin08], [Rie00].