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Marker D An Invitation To Mathematical Logic 2024
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In addition to covering the essentials, the author’s intention in writing this text is to entice the reader to further study mathematical logic. There is no current “standard text” for a first graduate course in mathematical logic and this book will fill that gap. While there is more material than could be covered in a traditional one semester course, an instructor can cover the basics and still have the flexibility to choose several weeks’ worth of interesting advanced topics that have been introduced. The text can and will be used by people in various courses with different sorts of perspectives. This versatility is one of the many appealing aspects of this book. A list of suggested portions to be covered in a single course is provided as well as a useful chart which maps chapter dependencies. Additionally, a motivated student will have ample material for further reading. New definitions, formalism, and syntax have been streamlined to engage thereader quickly into the heart of logic and to more sophisticated topics. Part I and Part IV center on foundational questions, while Part III establishes the fundamentals of computability. Part II develops model theory, highlighting the model theory of the fields of real and complex numbers. The interplay between logic and other areas of mathematics, notably algebra, number theory, and combinatorics, are illustrated in Chapters 5, 6, 8, 14, and 16. For most of the text, the only prerequisite is mathematical maturity. The material should be accessible to first year graduate students or advanced undergraduates in mathematics, graduate students in philosophy with a solid math background, or students in computer science who want a mathematical introduction to logic. Prior exposure to logic is helpful but not assumed. Introduction Detailed Overview Using This Book as a Text Prerequisites Acknowledgments Notation Chapter Dependencies Truth and Proof Languages, Structures, and Theories Languages Terms Formulas Satisfaction Normal Forms Negation Normal Form Disjunctive Normal Form The Coincidence Lemma Theories Logical Consequences Definable Sets Exercises Embeddings and Substructures Homomorphisms Embeddings and Substructures Isomorphism and Elementary Equivalence Elementary Embeddings Exercises Formal Proofs Exercises Gödel's Completeness Theorem Exercises Elements of Model Theory Compactness and Complete Theories Complete and κ-Categorical Theories Decidable Theories Transfer Results Back-and-Forth The Random Graph Exercises Ultraproducts Filters and Ultrafilters Ultraproducts Ultraproducts and Compactness Ultrapowers and Elementary Extensions Exercises Quantifier Elimination Diagrams Quantifier Elimination Tests Divisible Abelian Groups Ordered Divisible Abelian Groups Algebraically Closed Fields Definable and Constructible Sets The Nullstellensatz Exercises Model Theory of the Real Field Real Closed Fields Quantifier Elimination Semialgebraic Sets o-Minimal Expansions of R Ran and Subanalytic Sets Exponentiation Exercises Computability Models of Computation Register Machines Primitive Recursive Functions The Recursive Functions The Church–Turing Thesis Turing Machines Exercises Universal Machines and Undecidability Universal Machines The Halting Problem The Undecidability of Validity Index Sets The Recursion Theorem Exercises Computably Enumerable and Arithmetic Sets Computably Enumerable Sets Many-One Reducibility Computably Inseparable Sets Arithmetic Sets Complete Sets Kolmogorov Randomness Exercises Turing Reducibility Turing Reducibility and the Arithmetic Hierarchy Constructions in the Turing Degrees Incomparable Sets Inverting the Jump Minimal Degrees The Low Basis Theorem Post's Problem Exercises Arithmetic and Incompleteness Gödel's Incompleteness Theorems 1-Formulas Gödel's β-Function 1-Completeness of PA- The Representation Lemma Arithmetization of Syntax The Second Incompleteness Theorem Arithmetized Completeness The Second Incompleteness Theorem Revisited Exercises Hilbert's Tenth Problem Pell Equations Other Rings Exercises Peano Arithmetic and ε0 Goodstein's Sequences Ordinals
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