Classical computable model theory is most naturally concerned with countable domains. There are, however, several methods - some old, some new - that 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. This book contains introductions to eight major approaches to computable uncountable mathematics: descriptive set theory; infinite time Turing machines; Blum-Shub-Smale computability; Sigma-definability; computability theory on admissible ordinals; E-recursion theory; local computability; and uncountable reverse mathematics. This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages. It is ideal as both an introductory text for graduate and advanced undergraduate students, and a source of interesting new approaches for researchers in computability theory and related areas.--
Computable model theory is also related to reverse mathematics, the project of classifying theorems of mathematics ... There is no canonical generalization of this sort, and so the kind of effective theory of uncountable mathematics one ...
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... Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, and Jeff Tollaksen, The Mathematics of Superoscillations, 2017 C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Topologically Protected States in One-Dimensional Systems, ...
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We believe that Theorem [4] ought to be a corollary to a more general result connecting effectiveness properties of the 6 back-and-forth ... To appear in the ASL Lecture Notes in Logic volume Effective Mathematics of the Uncountable ...
Goncharov [122] and Manasse [230] gave examples of intrinsically c.e. relations on computable structures, which are not relatively intrinsically c.e. This result was lifted to higher levels in the hyperarithmetic hierarchy by Goncharov, ...
Since these structures are a routine part of the practice of pure and applied mathematics, a growing body of literature has addressed effective mathematics on uncountable structures (see, for instance [1,7,8,9,12,15,17,18]).