Forcing Iterated Ultrapowers And Turing Degrees
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Forcing, Iterated Ultrapowers, And Turing Degrees
This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2010 and 2011 Asian Initiative for Infinity Logic Summer Schools. The major topics covered set theory and recursion theory, with particular emphasis on forcing, inner model theory and Turing degrees, offering a wide overview of ideas and techniques introduced in contemporary research in the field of mathematical logic.
Reverse Mathematics
Author: Damir D. Dzhafarov
language: en
Publisher: Springer Nature
Release Date: 2022-07-25
Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.
New Frontiers in Science in the Era of AI
Author: Marilena Streit-Bianchi
language: en
Publisher: Springer Nature
Release Date: 2024-10-10
This interdisciplinary book enables scientists and non-specialists from various fields to delve into fascinating historical and recent scientific advancements in physics, astrophysics, genetic evolution, neuroscience, and artificial intelligence. Paradigm shifts are common in science, but some have significantly changed our perception and understanding of the world. This volume not only explores the profound implications of these scientific frontiers but also forecasts their impact on daily life. It delves into ongoing research and the technology that fuels advancements in physics and related fields. Authored by renowned researchers and science communicators, this book appeals to researchers and non-experts seeking a comprehensive overview of emerging developments that challenge ongoing research and will impact education and society in the coming years. Structured into distinct sections - New Physics World, Evolutionary Genetics to Epigenetics, Neurosciences and Neurophilosophy, Applications of Artificial Intelligence, and The Mystery of the Dark Sector - each chapter provides a deep dive into its respective subject, shedding light on the mysteries and implications of cutting-edge science. The book ends with an Epilogue illustrating the limitations of and challenges for our cognitive abilities.