An Invitation to Mathematical Logic

• Autore: Marker David
• Pubblicazione: Cham : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (359 pages)
• Nota di contenuto: Intro -- Introduction -- Detailed Overview -- Using This Book as a Text -- Prerequisites -- Acknowledgments -- Notation -- Chapter Dependencies -- Contents -- Part I Truth and Proof -- 1 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 -- 2 Embeddings and Substructures -- Homomorphisms -- Embeddings and Substructures -- Isomorphism and Elementary Equivalence -- Elementary Embeddings -- Exercises -- 3 Formal Proofs -- Exercises -- 4 Gödel's Completeness Theorem -- Exercises -- Part II Elements of Model Theory -- 5 Compactness and Complete Theories -- Complete and κ-Categorical Theories -- Decidable Theories -- Transfer Results -- Back-and-Forth -- The Random Graph -- Exercises -- 6 Ultraproducts -- Filters and Ultrafilters -- Ultraproducts -- Ultraproducts and Compactness -- Ultrapowers and Elementary Extensions -- Exercises -- 7 Quantifier Elimination -- Diagrams -- Quantifier Elimination Tests -- Divisible Abelian Groups -- Ordered Divisible Abelian Groups -- Algebraically Closed Fields -- Definable and Constructible Sets -- The Nullstellensatz -- Exercises -- 8 Model Theory of the Real Field -- Real Closed Fields -- Quantifier Elimination -- Semialgebraic Sets -- o-Minimal Expansions of R -- Ran and Subanalytic Sets -- Exponentiation -- Exercises -- Part III Computability -- 9 Models of Computation -- Register Machines -- Primitive Recursive Functions -- The Recursive Functions -- The Church-Turing Thesis -- Turing Machines -- Exercises -- 10 Universal Machines and Undecidability -- Universal Machines -- The Halting Problem -- The Undecidability of Validity -- Index Sets -- The Recursion Theorem -- Exercises. ; 11 Computably Enumerable and Arithmetic Sets -- Computably Enumerable Sets -- Many-One Reducibility -- Computably Inseparable Sets -- Arithmetic Sets -- Complete Sets -- Kolmogorov Randomness -- Exercises -- 12 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 -- Part IV Arithmetic and Incompleteness -- 13 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 -- 14 Hilbert's Tenth Problem -- Pell Equations -- Other Rings -- Exercises -- 15 Peano Arithmetic and ε0 -- Goodstein's Sequences -- Ordinals < -- ε0 -- Hardy Functions -- Infinitary Proof Theory for PA -- Our Plan -- Bounded Deductions -- Embedding Proofs from PA+ -- Toward Cut Elimination -- Cut Elimination -- Bounds on Growth -- Consistency of PA -- Exercises -- 16 Models of Arithmetic and Independence Results -- Ramsey Theory -- Independence of the Paris-Harrington Principle -- Diagonal Indiscernibles -- Models of PA -- End Extensions -- Cofinal Extensions -- Standard Systems -- Friedman's Embedding Theorem -- Exercises -- A Set Theory -- Axioms for Set Theory -- Well Orderings and Ordinals -- Cardinals -- The Axiom of Choice -- Cardinal Arithmetic -- B Unique Readability -- Unique Readability of Terms -- Unique Readability of Formulas -- C Real Algebra -- Bibliography -- Index.
• Titolo autorizzato: An Invitation to Mathematical Logic
• ISBN: 3-031-55368-3
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910855373203321
• Lo trovi qui: Univ. Federico II
• Serie: Graduate Texts in Mathematics Series