Computable Analysis Over the Generalized Baire Space MSc Thesis (Afstudeerscriptie) written by Lorenzo Galeotti (born May 6th, 1987 in Viterbo, Italy) under the supervision of Prof. Dr. Benedikt L¨owe and Drs. Hugo Nobrega, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: July 21st, 2015 Dr. Alexandru Baltag (Chair) Dr. Benno van den Berg Dr. Yurii Khomskii Prof. Dr. Benedikt L¨owe Drs. Hugo Nobrega Dr. Arno Pauly Dr. Benjamin Rin Contents 1 Introduction 1 2 Basics 4 2.1 Orders, Fields and Topology . .4 2.2 Groups and Fields Completion . .7 2.3 Surreal Numbers . 10 2.3.1 Basic Definitions . 10 2.3.2 Operations Over No . 13 2.3.3 Real Numbers and Ordinals . 15 2.3.4 Normal Form . 16 2.4 Baire Space and Generalized Baire Space . 17 2.5 Computable Analysis . 19 2.5.1 Effective Topologies and Representations . 19 2.5.2 Subspaces, Products and Continuous Functions . 21 2.5.3 The Weihrauch Hierarchy . 22 3 Generalizing R 24 3.1 Completeness and Connectedness of Rκ .............................. 24 3.2 κ-Topologies . 26 3.3 Analysis Over Super Dense κ-real Extensions of R ........................ 29 3.4 The Real Closed Field Rκ ...................................... 32 3.5 Generalized Descriptive Set Theory . 42 4 Generalized Computable Analysis 52 4.1 Wadge Strategies . 52 4.2 Computable Analysis Over κκ .................................... 54 4.3 Restrictions, Products and Continuous Functions Representations . 58 4.4 Representations for Rκ ........................................ 63 4.5 Generalized Choice Principles . 68 4.6 Baire Choice Functions . 76 4.7 Representation of the IVT . 79 5 Conclusions and Open Questions 85 5.1 Summary . 85 5.2 Future Work . 86 5.3 Open Questions . 87 1 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 !! to any space of cardinality 2@0 , 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@0 , 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 κκ for cardinals κ > ! 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 com- putable analysis to generalized Baire spaces. We will introduce Rκ, a Cauchy-complete real closed field of κ cardinality 2 with κ uncountable. We will prove that Rκ 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 Rκ. We shall show that Rκ is a good candidate for extending computable analysis to the generalized Baire κ space κ . In particular, we generalize many of the most important representations of R to Rκ and we show that these representations are well-behaved with respect to the interval topology over Rκ. 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 Rκ from a computable analysis prospective. Chapter 1 Introduction Computable Analysis Computable analysis is the study of the computational properties of real analysis. We refer the reader to [28] and [6] for an introduction to classical computable analysis. In classical computability theory one studies the properties of functions over natural numbers and then transfers these properties to arbitrary countable spaces via coding. The same approach is taken in computable analysis. One of the main tools of computable analysis is the Baire space !!, namely the space of sequence of natural numbers of length !. Following the classical computability theory approach, computational and topological properties of !! are studied and then transferred to spaces of cardinality 2@0 via coding. Coding R o !! Of particular interest in computable analysis is the study of the computational and topological content of theorems from classical analysis. The idea is that of formalizing the complexity of theorems by means similar to those used in computability theory to classify functions over the natural numbers. In this context, the Weihrauch theory of reducibility plays a predominant role. For an introduction to the theory of Weihrauch reductions see [5]. Weirauch reductions can be used to classify functions over the Baire space !!. Intuitively, a function f : !! ! !! is said to be Weihrauch reducible to g : !! ! !! if there are two continuous functions which translate f into g as shown in the following commuting diagram: Input Translation !! / !! f g Output Translation !! o !! Many theorems from classical analysis can be stated as formulas of the type: 8x 2 X9y 2 Y: '(x; y); with ' a quantifier-free formula. These formulas can be formalized by using multi-valued functions. A multi-valued function T : X ⇒ Y is a function that given an element x of X returns a subset of Y . Let us consider two classical examples, namely the Intermediate Value Theorem and the Baire Category Theorem. The statement of the Intermediate Value Theorem is the following: For every continuous function f :[a; b] ! R such that f(a) · f(b) < 0 there is a real number c 2 [a; b] such that f(c) = 0. Therefore it can be stated as follows: 8f 2 C[a;b]9c 2 [a; b]: f(c) = 0; 1 where C[a;b] is the set of continuous functions f :[a; b] ! R such that f(a) · f(b) < 0. We can formalize this formula by the following multi valued function: IVT : C[a;b] ⇒ [a; b]; where, given a function f 2 C[a;b], the set IVT(f) ⊂ [a; b] is such that c 2 IVT(f) ) f(c) = 0: The Baire Category Theorem can be stated as follows: Given a countable sequence of closed nowhere dense subsets (An)n2! of a complete separable metric space S X, the set X n n2! An is not empty. Therefore it can be formalized by the following multi valued function: BCT : A(X)N ⇒ X; where A(X)N is the set of the countable sequences of closed nowhere dense subsets of X. Given a sequence (An)n2!, we have that: [ BCT((An)n2!) 2 X n An: n2! The previous two examples show that even though both the Intermediate Value Theorem and the Baire Category Theorem have a similar logical form, the multi-valued functions that represent them are quite different. It seems then really impractical to compare these two multi-valued functions directly. This apparent difficulty can be overcome by using the Baire space. A multi-valued function T : X ⇒ Y is usually coded within the Baire space as the set of functions t : ! ! ! such that for every p 2 !!, we have that C(f(p)) 2 T (C(p)) where C is the function coding X ! in ! . Given two multi-valued functions T1 : X1 ⇒ Y1 and T2 : X2 ⇒ Y2 one can therefore compare their complexity by studying the Weihrauch reducibility of their codings. In particular, one can study what is the relationship, with respect to Weihrauch reducibility, of the representations of T1 and T2. For this reason it is natural to use the Weihrauch theory of reducibility to compare theorems from analysis. The following diagram illustrates the situation for IVT and BCT: Translation Coding ! + ! Coding C[a;b] o ! k ! / A(X)N Translation IVT ivt bct BCT Translation [a; b] o !! + !! / X Coding k Coding Translation By using this technique it is possible to arrange many theorems from classical real analysis in a complexity hierarchy called the Weihrauch hierarchy. A study of the Weihrauch degrees of the most important theorems from real analysis can be found in [5] and [2]. Generalized Baire Spaces Recently, generalizations of the Baire space to uncountable cardinals have been of great interest for descriptive set theorists. We refer the reader to [13] for an introduction to generalized descriptive set theory. Even though the theory of generalized Baire spaces κκ with κ uncountable is not a new concept in set theory, many aspects of this theory are still unknown. In particular it is still unclear how these generalizations can be used in the context of computable analysis. In this thesis we will begin for the first time the study of generalized computable analysis, namely the generalization of computable analysis to generalized Baire spaces. Given a space M of cardinality 2κ, the idea is that of substituting the Baire space !! with the generalized Baire space κκ and then of developing the machinery necessary in order to transfer topological properties form κκ to M. In particular we will be interested in the study of the Weihrauch hierarchy in the context of generalized Baire spaces. Since in classical computable analysis and classical Weihrauch theory the field of real numbers has a central role, a question arises naturally: 2 What is the right generalization of R in the context of generalized computable analysis? Coding R o !! Generalization Generalization Coding ? o κκ One of the main results of this thesis is the definition of Rκ, a generalization of the real line which provides a well-behaved environment for generalizing real analysis and for developing generalized computable analysis.