§1. Introduction. Turing computability has always been restricted to maps on countable sets. This restriction is inherent in the nature of a Turing machine: a computation is performed in a finite length of time, so that even if the available input was a countable binary sequence, only a finite initial segment of that sequence was actually used in the computation. The Use Principle then says that an input of any other infinite sequence with that same initial segment will result in the same computation and the same output. Thus, while the domain might have been viewed as the (uncountable) set of infinite binary sequences, the countable domain containing all finite initial segments would have sufficed.

To be sure, there are approaches that have defined natural notions of computable functions on uncountable sets. The bitmap model, detailed in [3] and widely used in computable analysis, is an excellent model for computability on Cantor space 2ω. On the real numbers ℝ, however, it fails to compute even the simplest discontinuous functions, which somewhat limits its utility. The Blum-Shub-Smale model (see [2]) expands the set of functions which we presuppose to be computable. Having done so, it gives an elegant account of computable functions on the reals, with nice analogies to computability on ω, but the initial assumption immediately distances it from Turing's original concept of computability.

§1. Introduction. The enterprise of calibrating the strength of theorems of classical mathematics in terms of the (set existence) axioms needed to prove them, was begun by Harvey Friedman in the 1970s (as in [6] and [7]). It is now called Reverse Mathematics as, to prove that some set of axioms is actually necessary to establish a given theorem, one reverses the standard paradigm by proving that the axioms follow from the theorem (in some weak base theory). The original motivations for the subject were foundational and philosophical. It has become a remarkably fruitful and successful endeavor supplying a framework for both the philosophical questions about existence assumptions and foundational or mathematical ones about construction techniques needed to actually produce the objects that the theorems assert exist.

§1. Introduction. When looking at structures of size the continuum from an effective viewpoint, the following definition is a natural generalization of ideas from computable model theory.

Definition 1.1. Let X be either 2ω, ωω or ℝ, and let C be a (complexity) class of relations on X. A C-presentation of a structure A is a tuple of relations S = (D, E, R1,…, Rn) such that

о All D, E, R1,…, Rn are in C;

о D ⊆ X and E is an equivalence relation on D (D is called the domain);

о R1, …, Rn are relations compatible with E.

S is a C-representation of A if A ≅ S/E. When E is the identity on D, we say that S is an injective C-presentation of A.

There are various possible choices for C. In this paper we concentrate on the case that C is the class of Borel relations. Given a topological space X as above, the σ-algebra of Borel sets is the smallest σ-algebra containing the open sets.

§1. Introduction. Our aim is to develop computable structure theory for uncountable structures. In this paper we focus on structures of size ℵ1. The fundamental decision to be made, when trying to formulate such a theory, is the choice of computability tools that we intend to use. To discover which structures are computable, we need to first describe which subsets of the domain are computable, and which functions are computable. In this paper, we use admissible recursion theory (also known as α-recursion theory) over the domain ω1. We believe that this choice yields an interesting computable structure theory. It also illuminates the concepts and techniques of classical computable structure theory by observing similarities and differences between the countable and uncountable settings. In particular, it seems that as is the case for degree theory and for the study of the lattice of c.e. sets, the difference between true finiteness and its analogue in the generalised case, namely countability in our case, is fundamental to some constructions and reveals a deep gap between classical computability and attempts to generalise it to the realm of the uncountable.

Chang and Keisler [8] famously defined model theory as the sum of logic and universal algebra. In the same spirit, one might describe computable model theory to be the investigation of the constraints on information content imposed by algebraic structure. The analogue of the interplay between syntactical objects and the algebraic structure they deine is the connection between deinability and complexity. One asks: How complicated are the constructions of model theory and algebra? What kind of information can be coded in structures like groups, ields, graphs, and orders? What mathematical distinctions are unearthed when “boldface” notions such as isomorphism are replaced by their “lightface” analogues such as, say, computable isomorphism?

A special case of the following definition was first rigorously made by Fröhlich and Shepherdson [11], following work of Hermann [17] and van der Waerden [40], which itself built on the constructive tradition of 19th century algebra. It was further developed by Rabin [32, 33] and Mal'cev [27].

Definition. Let ℒ be a computable signature (language), and let ℳ be an ℒ-structure whose universe is the set of natural numbers. The degree of ℳ is the Turing degree of the atomic (equivalently, quantifier-free) diagram of ℳ.

A structure is computable if its degree is 0, the Turing degree of computable sets. Equivalently, a structure ℳ is computable if, uniformly in the symbols of ℒ, the interpretations in ℳ of the constant symbols, function symbols, and relation symbols of ℒ are computable.

§1. Introduction. The theory of effectiveness properties on countable structures whose atomic diagrams are Turing computable is well-studied (see, for instance, [1, 15]). Typical results describe which structures in various classes are computable (or have isomorphic copies that are) [19], or the potential degree of unsolvability of various definable subsets of the structure [16]. The goal of the present paper is to survey some initial results investigating similar concerns on structures which are effective in a different sense.

A rather severe limitation of the Turing model of computability is its traditional restriction to the countable. Of course, many successful generalizations have been made (see, for instance, [28, 12, 13, 23, 24, 26] and the other chapters in the present volume). The generalization that will be treated here is based on the observation that while there is obviously no Turing machine for addition and multiplication of real numbers, there is strong intuition that these operations are “computable.” The BSS model of computation, first introduced in [5], approximately takes this to be the definition of computation on a given ring (a more formal definition is forthcoming). This allows several problems of computation in numerical analysis and continuous geometry to be treated rigorously. The monograph [4] gives the examples of the “decision problem” of the points for which Newton's method will converge to a root, and determining whether a given point is in the Mandelbrot set.

Although classical computable model theory is most naturally concerned with countable domains, several methods – some old, some new – have extended its basic concepts to uncountable structures. Unlike in the classical case, however, no single dominant approach has emerged, and different methods reveal different aspects of the computable content of uncountable mathematics. Furthermore, uncountable computable model theory is still in an early stage of development, and, in particular, there has been relatively little work on connecting and comparing the various available approaches. Two Effective Mathematics of the Uncountable workshops were held at the CUNY Graduate Center in New York on August 18–22, 2008 and August 17–21, 2009, organized by Noam Greenberg, Joel Hamkins, Denis Hirschfeldt, and Russell Miller, with support from a Templeton Foundation “Exploring the Infinite” program grant. The aim of these workshops was to introduce a variety of approaches to uncountable computable model theory to researchers and students in computability theory and related fields, and to encourage collaboration between those who have developed and studied different facets of the effective content of uncountable mathematics.

Speaking at the EMU workshops were researchers with a wide range of backgrounds and motivations: Nate Ackerman, Wesley Calvert, Samuel Coskey, Noam Greenberg, Joel Hamkins, Denis Hirschfeldt, Julia Knight, Peter Koepke, David Linetsky, Robert Lubarsky, Russell Miller, Antonio Montalbán, Ansten Mørch Klev, Kerry Ojakian, Gerald Sacks, Richard Shore, Alexei Stukachev, and Philip Welch.