Computable Analysis Over the Generalized Baire Space
Lorenzo Galeotti
Abstract:
One of the main goals of computable analysis is that of formalizing the complexity of theorems from real analysis. In this setting Weihrauch reductions play the role that Turing reductions do in standard computability theory. Via coding, we can transfer computability and topological results from the Baire space ω^ω to any space of cardinality 2^א0 , so that e.g. functions over R can be coded as functions over the Baire space and then studied by means of Weihrauch reductions. Since many theorems from analysis can be thought to as functions between spaces of cardinality 2^א0 , computable analysis can then be used to study their complexity and to order them in a hierarchy.
Recently, the study of the descriptive set theory of the generalized Baire spaces κ^κ for cardinals κ > ω has been catching the interest of set theorists. It is then natural to ask if these generalizations can be used in the context of computable analysis.
In this thesis we start the study of generalized computable analysis, namely the generalization of computable analysis to generalized Baire spaces. We will introduce R_κ , a Cauchy-complete real closed field of cardinality 2^κ with κ uncountable. We will prove that R_κ shares many features with R which have a key role in real analysis. In particular, we will prove that a restricted version of the intermediate value theorem and of the extreme value theorem hold in R_κ.
We shall show that R_κ is a good candidate for extending computable analysis to the generalized Baire κspace κ. In particular, we generalize many of the most important representations of R to R_κ and we show that these representations are well-behaved with respect to the interval topology over R_κ.
In the last part of the thesis, we begin the study of the Weihrauch hierarchy in this generalized context. We generalize some of the most important choice principles which in the classical case characterize the Weihrauch hierarchy. Then we prove that some of the classical Weihrauch reductions can be extended to these generalizations. Finally we will start the study of the restricted version of the intermediate value theorem which holds for R_κ from a computable analysis prospective.