Categoricity
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Categoricity
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Author : John T. Baldwin
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Categoricity written by John T. Baldwin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
"Modern model theory began with Morley's categoricity theorem: A countable first-order theory that has a unique (up to isomorphism) model in one uncountable cardinal (i.e., is categorical in cardinality) if and only if the same holds in all uncountable cardinals. Over the last 35 years Shelah made great strides in extending this result to infinitary logic, where the basic tool of compactness fails. He invented the notion of an Abstract Elementary Class to give a unifying semantic account of theories in first-order, infinitary logic and with some generalized quantifiers. Zilber developed similar techniques of infinitary model theory to study complex exponentiation." "This book provides the first unified and systematic exposition of this work. The many examples stretch from pure model theory to module theory and covers of Abelian varieties. Assuming only a first course in model theory, the book expounds eventual categoricity results (for classes with amalgamation) and categoricity in excellent classes. Such crucial tools as Ehrenfeucht-Mostowski models, Galois types, tameness, omitting-types theorems, multi-dimensional amalgamation, atomic types, good sets, weak diamonds, and excellent classes are developed completely and methodically. The (occasional) reliance on extensions of basic set theory is clearly laid out. The book concludes with a set of open problems." --Book Jacket.
Philosophical Uses Of Categoricity Arguments
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Author : Penelope Maddy
language : en
Publisher: Cambridge University Press
Release Date : 2023-12-21
Philosophical Uses Of Categoricity Arguments written by Penelope Maddy 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 2023-12-21 with Philosophy categories.
This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw – the same theorem might successfully support one such conclusion while failing to support another. It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. Highlighting the roles of first- and second-order theorems, of external and internal theorems, the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.
Uncountably Categorical Theories
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Author : Boris Zilber
language : en
Publisher: American Mathematical Soc.
Release Date :
Uncountably Categorical Theories written by Boris Zilber and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
The 1970s saw the appearance and development in categoricity theory of a tendency to focus on the study and description of uncountably categorical theories in various special classes defined by natural algebraic or syntactic conditions. There have thus been studies of uncountably categorical theories of groups and rings, theories of a one-place function, universal theories of semigroups, quasivarieties categorical in infinite powers, and Horn theories. In Uncountably Categorical Theories , this research area is referred to as the special classification theory of categoricity. Zilber's goal is to develop a structural theory of categoricity, using methods and results of the special classification theory, and to construct on this basis a foundation for a general classification theory of categoricity, that is, a theory aimed at describing large classes of uncountably categorical structures not restricted by any syntactic or algebraic conditions.
Definability And Categoricity In Continuous Logic
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Author : James Ernest Hanson
language : en
Publisher:
Release Date : 2020
Definability And Categoricity In Continuous Logic written by James Ernest Hanson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.
After a self-contained development of continuous first-order logic, we study the phenomena of definability and categoricity in continuous logic. The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. We investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end, we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out, and we show that [omega]-stable theories and randomizations of arbitrary continuous theories are dictionaric. We also introduce, in the context of Banach theories, 'indiscernible subspaces,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable outside of the context of inseparably categorical theories. We construct or present a slew of counterexamples, including an [omega]-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension. In order to investigate notions of approximate categoricity, we give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of Ben Yaacov and Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. We introduce distortion systems, which are a mild generalization of perturbation systems. With this we explicitly exhibit Scott sentences for perturbation systems, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically, by distortion systems, and semantically, by certain elementary classes of two-sorted structures that witness approximate isomorphism. We also make progress towards an analog of Morley's theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal [kappa] such that every model of size [kappa] is 'approximately saturated,' in the appropriate sense, then the same is true for all uncountable cardinalities. Finally, we present some non-trivial examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.
On Morley S Categoricity Theorem With An Eye Toward Forking
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Author : Colin N. Craft
language : en
Publisher:
Release Date : 2011
On Morley S Categoricity Theorem With An Eye Toward Forking written by Colin N. Craft and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Model theory categories.
The primary result of this paper is Morley's Categoricity Theorem that a complete theory T which is k-catecorigal for some uncountable cardinal k is ^-categorical for every uncountable cardinal ^. We prove this by proving a characterization of uncountably categorical theories due to Baldwin and Lachlan. Before the actual statement and proof of Morley's theorem, we give an overview of the prerequisites from mathematical logic needed to understand the theorem and its proof. After proving Morley's theorem we briefly indicate some possible directions of further study having to do with forking and the related notion of independence of types.
Degrees Of Categoricity And The Isomorphism Problem
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Author : Mohammad Assem Abdalqader Mahmoud
language : en
Publisher:
Release Date : 2019
Degrees Of Categoricity And The Isomorphism Problem written by Mohammad Assem Abdalqader Mahmoud and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Categories (Mathematics) categories.
In this thesis, we study notions of complexity related to computable structures. We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal $\alpha$, there exists a computable tree of rank $\alpha+1$ with strong degree of categoricity ${\bf 0}^{(2\alpha)}$ if $\alpha$ is finite, and with strong degree of categoricity ${\bf 0}^{(2\alpha+1)}$ if $\alpha$ is infinite. For a computable limit ordinal $\alpha$, we show that there is a computable tree of rank $\alpha$ with strong degree of categoricity ${\bf 0}^{(\alpha)}$ (which equals ${\bf 0}^{(2\alpha)}$). In general, it is not the case that every Turing degree is the degree of categoricity of some structure. However, it is known that every degree that is of a computably enumerable (c.e.) set\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a successor ordinal, is a degree of categoricity. In this thesis, we include joint work with Csima, Deveau and Harrison-Trainor which shows that every degree c.e.\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e.\ in and above $\mathbf{0}^{(\omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk. After that, we study the isomorphism problem for tree structures. It follows from our proofs regarding the degrees of categoricity for these structures that, for every computable ordinal $\alpha>0$, the isomorphism problem for trees of rank $\alpha$ is $\Pi_{2\alpha}$-complete. We also discuss the isomorphism problem for pregeometries in which dependent elements are dense and the closure operator is relatively intrinsically computably enumerable. We show that, if $K$ is a class of such pregeometries, then the isomorphism problem for the class $K$ is $\Pi_3$-hard. Finally, we study the Turing ordinal. We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal. We give examples of classes of computable structures such that the upper Turing ordinal is $\beta$ and the lower Turing ordinal is $\alpha$ for all computable ordinals $\alpha
Polynomial Time Categoricity And Linear Orderings
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Author : Jeffrey B. Remmel
language : en
Publisher:
Release Date : 1993
Polynomial Time Categoricity And Linear Orderings written by Jeffrey B. Remmel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with categories.
Proceedings Of The 14th And 15th Asian Logic Conferences
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Author : Kim Byunghan
language : en
Publisher: World Scientific
Release Date : 2019-01-16
Proceedings Of The 14th And 15th Asian Logic Conferences written by Kim Byunghan and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-01-16 with Mathematics categories.
The Asian Logic Conference (ALC) is a major international event in mathematical logic. It features the latest scientific developments in the fields of mathematical logic and its applications, logic in computer science, and philosophical logic. The ALC series also aims to promote mathematical logic in the Asia-Pacific region and to bring logicians together both from within Asia and elsewhere for an exchange of information and ideas. This combined proceedings volume represents works presented or arising from the 14th and 15th ALCs.
An Analysis Of The Concept Of Constructive Categoricity
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Author : Charles Francis Quinn
language : en
Publisher:
Release Date : 1971
An Analysis Of The Concept Of Constructive Categoricity written by Charles Francis Quinn and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971 with Categories (Philosophy) categories.
Model Theory And The Philosophy Of Mathematical Practice
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Author : John T. Baldwin
language : en
Publisher: Cambridge University Press
Release Date : 2018-01-25
Model Theory And The Philosophy Of Mathematical Practice written by John T. Baldwin 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 2018-01-25 with Mathematics categories.
Recounts the modern transformation of model theory and its effects on the philosophy of mathematics and mathematical practice.