Seminar room of the IMUB. Universitat de Barcelona, Gran Via de les Corts Catalanes, 585.
Abstract
Although ordinals appear in many branches of mathematics, they have a particularly prominent role in proof theory, and this in turn has produced a wealth of notation systems for rather large countable ordinals. A recent paradigm in proof theory is based on provability algebras, which are structures that capture the behavior of Gödels provability predicate in arithmetic. One nice thing about such algebras is that familiar proof-theoretic ordinals may be found naturally within them, forming a sort of "backbone".
In this talk we shall sketch how these ordinals appear, beginning with the ordinal ε0 of Peano Arithmetic, through the Feferman-Shütte ordinal which arises through transfinite (but countable) proofs, and finally to much larger ordinals related to uncountable proofs, such as the Howard-Bachmann ordinal. The end result will be an ordinal notation system which extends Beklemishev's "brackets" representation of the
Feferman-Schütte ordinal into the realm of impredicativity.
Supported by:
Centre de Recerca Matemàtica (CRM)
Department of Applied Mathematics II of the Universitat Politècnica
de Catalunya
Department of Logic, History and Philosophy of Science of the
University of Barcelona
Department of Philosophy of the Universitat Autònoma de Barcelona
Department of Probability, Logic and Statistics of the
University of Barcelona
Institut de Matemàtica de la Universitat de Barcelona (IMUB)
