Computability on the Countable Ordinals and the Hausdorff-Kuratowski Theorem (Extended Abstract)

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of countable ordinals via a representation in the sense of computable analysis. The computability structure is characterized by the computability of four specific operations, and we prove further relevant operations to be computable. Some alternative approaches are discussed, too. As an application in effective descriptive set theory, we can then state and prove computable uniform versions of the Lusin separation theorem and the Hausdorff-Kuratowski theorem. Furthermore, we introduce an operator on the Weihrauch lattice corresponding to iteration of some principle over a countable ordinal

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Three models of ordinal computability

In this thesis we expand the scope of ordinal computability, i.e., the study of models of computation that are generalized to infinite domains. The discipline sets itself apart from classical work on generalized recursion theory by focusing strongly on the computational paradigm and an analysis in elementary computational steps. In the present work, two models of classical computability of which no previous generalizations to ordinals are known to the author are lifted to the ordinal domain, namely λ-calculus and Blum-Shub-Smale machines. One of the multiple generalizations of a third model relevant to this thesis, the Turing machine, is employed to further study classical descriptive set theory. The main results are: An ordinal λ-calculus is defined and confluency properties in the form of a weak Church-Rosser theorem are established. The calculus is proved to be strongly related to the constructible hierarchy of sets, a feature typical for an entire subfamily of models of ordinal computation. Ordinal Turing machines with input restricted to subsets of ω are shown to compute the Δ12 sets of reals. Conversely, the machines can be employed to reprove the absoluteness of Σ12 sets (Shoenfield absoluteness) and basic properties of Σ12 sets. New tree representations and new pointclasses defined by the means of ordinal Turing computations are introduced and studied. The Blum-Shub-Smale model for computation on the real numbers is lifted to transfinite running times. The supremum of possible runtimes is determined and an upper bound on the computational strength is given. Summarizing, this thesis both expands the field of ordinal computability by enlarging its palette of computational models and also connects the field further by tying in the new models into the existing framework. Questions that have been raised in the community, e.g. on the possibility of generalizations of λ-calculus and Blum-Shub-Smale machines, are addressed and answered