Ultraproducts Of O Minimal Structures

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Ultraproducts Of O Minimal Structures

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Author : Alex Rennet
language : en
Publisher:
Release Date : 2012

Ultraproducts Of O Minimal Structures written by Alex Rennet and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.

There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures. In the first part we investigate (for a fixed first-order language L) what we call the L-theory of o-minimality. It is the theory consisting of those L-sentences true in all o-minimal L-structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L-theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list of axioms A which is consistent with the L-theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal. In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting. In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.

Finite And Algorithmic Model Theory

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Author : Javier Esparza
language : en
Publisher: Cambridge University Press
Release Date : 2011-03-10

Finite And Algorithmic Model Theory written by Javier Esparza and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-03-10 with Computers categories.

Surveys of current research in logical aspects of computer science that apply finite and infinite model-theoretic methods.

A Shorter Model Theory

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Author : Wilfrid Hodges
language : en
Publisher: Cambridge University Press
Release Date : 1997-04-10

A Shorter Model Theory written by Wilfrid Hodges and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-04-10 with Mathematics categories.

This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory.

Logic Colloquium 99

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Author : Jan Van Eijck
language : en
Publisher: Cambridge University Press
Release Date : 2017-03-30

Logic Colloquium 99 written by Jan Van Eijck and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-30 with Mathematics categories.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the seventeenth publication in the Lecture Notes in Logic series, collects the proceedings of the European Summer Meeting of the Association for Symbolic Logic, held in Utrecht, The Netherlands in August, 1999. It includes surveys and research articles from some of the world's preeminent logicians. Two long articles are based on tutorials given at the meeting and present accessible expositions of current research in geometric model theory and the descriptive set theory of group actions. The other articles cover current research topics in all areas of mathematical logic, including proof theory, set theory, model theory, computability theory and philosophy.

Representation And Productive Ambiguity In Mathematics And The Sciences

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Author : Emily R. Grosholz
language : en
Publisher: Clarendon Press
Release Date : 2007-08-30

Representation And Productive Ambiguity In Mathematics And The Sciences written by Emily R. Grosholz and has been published by Clarendon Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-08-30 with Mathematics categories.

Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian 'analysis' - the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience. Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable for mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are sublty altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

The Use Of Ultraproducts In Commutative Algebra

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Author : Hans Schoutens
language : en
Publisher: Springer
Release Date : 2010-07-16

The Use Of Ultraproducts In Commutative Algebra written by Hans Schoutens and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-16 with Mathematics categories.

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

Proceedings Of The International Congress Of Mathematicians 2018 Icm 2018 In 4 Volumes

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Author : Boyan Sirakov
language : en
Publisher: World Scientific
Release Date : 2019-02-27

Proceedings Of The International Congress Of Mathematicians 2018 Icm 2018 In 4 Volumes written by Boyan Sirakov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-27 with Mathematics categories.

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Mathematical Reviews

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Author :
language : en
Publisher:
Release Date : 2003

Mathematical Reviews written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.

Foundations Of Formal Logic

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Author : Richard Johnson
language : en
Publisher: HiTeX Press
Release Date : 2025-06-12

Foundations Of Formal Logic written by Richard Johnson and has been published by HiTeX Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-06-12 with Computers categories.

"Foundations of Formal Logic" "Foundations of Formal Logic" presents a sweeping and rigorous exploration of the principles, systems, and philosophical underpinnings of modern formal logic. Beginning with a rich historical and philosophical context, the book traces the emergence of logical reasoning from its ancient origins through the revolutionary contributions of Aristotle and Frege, ultimately charting the evolution of formal systems and their foundational role in mathematics. Engaging with pivotal philosophical debates—spanning realism, nominalism, logicism, formalism, and intuitionism—it offers critical insight into the shifting landscape of logic’s interpretation and its continuing influence on contemporary thought. Delving into the technical heart of the subject, the text systematically covers the syntax and semantics of formal languages, propositional and predicate logic, and the intricate structures of model theory and proof theory. Readers are guided through the construction of formal languages, the architecture of proofs, and the profound implications of soundness, completeness, and compactness theorems. Advanced and non-classical logics—including modal, temporal, intuitionistic, paraconsistent, and fuzzy logics—are deftly introduced, highlighting the depth and versatility of formal reasoning in addressing complex phenomena. The latter chapters bridge logic with the wider intellectual landscape, illuminating its centrality to computer science, artificial intelligence, mathematics, linguistics, and the sciences. Foundational milestones, such as Gödel’s incompleteness theorems and Turing’s theory of computation, stand alongside discussions of formal verification, knowledge representation, and the logical foundations of databases. Concluding with a survey of open problems, new paradigms, and ethical dimensions, "Foundations of Formal Logic" equips readers with both a comprehensive reference and a roadmap to the enduring questions and future frontiers of logical inquiry.

Encyclopedia Of Philosophy

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Author : Donald M. Borchert
language : en
Publisher: Macmillan Reference USA
Release Date : 2006

Encyclopedia Of Philosophy written by Donald M. Borchert and has been published by Macmillan Reference USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Philosophy categories.

This volume, covering entries from "Masaryk, Tomas Garrigue" to "Nussbaum, Martha," presents articles on Eastern and Western philosophies, medical and scientific ethics, the Holocaust, terrorism, censorship, biographical entries, and much more.