Mathematical LogicCover -- Title Page -- Copyright Page -- PREFACE -- Table of Contents -- Dedication -- Chapter 1: The Nature of Mathematical Logic -- 1.1 Axiom systems -- 1.2 Formal systems -- 1.3 Syntactical variables -- Chapter 2: First-Order Theories -- 2.1 Functions and predicates -- 2.2 Truth functions -- 2.3 Variables and quantifiers -- 2.4 First-order languages -- 2.6 Logical axioms and rules -- 2.5 Structures -- Problems -- Chapter 3: Theorems in First-Order Theories -- 3.1 The tautology theorem -- 3.2 Results on quantifiers -- 3.3 The deduction theorem -- 3.4 The equivalence and equality theorems -- 3.5 Prenex form -- Problems -- Chapter 4: The Characterization Problem -- 4.1 The reduction theorem -- 4.2 The completeness theorem -- 4.3 The consistency theorem -- 4.4 Herbrand's theorem -- 4.5 Addition of function symbols -- 4.6 Extensions by definitions -- 4.7 Interpretations -- Problems -- Chapter 5: The Theory of Models -- 5.1 The compactness theorem -- 5.2 Isomorphisms and substructures -- 5.3 Cardinality of models -- 5.4 Joint consistency -- 5.5 Complete theories -- 5.6 Categoricity -- Problems -- Chapter 6: Incompleteness and Undecidability -- 6.1 Calculability -- 6.2 Recursive functions -- 6.3 Explicit definitions -- 6.4 Sequence numbers -- 6.5 Church's thesis -- 6.6 Expression numbers -- 6.7 Representability -- 6.8 Church's theorem and the incompleteness theorem -- 6.9 Undecidability -- Chapter 7: Recursion Theory -- 7.1 Partial functions -- 7.2 Functionals and relations -- 7.3 Properties of recursive functionals -- 7.4 Indices -- 7.5 The arithmetical hierarchy -- 7.6 Relative recursiveness -- 7.7 Degrees -- 7.8 The analytical hierarchy -- 7.9 Hyperarithmetical relations -- 7.10 The characterization theorem -- 7.11 Basis theorems -- Problems -- Chapter 8: The Natural Numbers -- 8.1 Peano arithmetic -- 8.2 The theorem on consistency proofs |
Contents
FirstOrder Theories | 9 |
vi | 15 |
Theorems in FirstOrder Theories | 26 |
Copyright | |
14 other sections not shown
Common terms and phrases
a₁ axiom of constructibility B₁ benign calculable Card(x cardinal closed formula consistency proof constructible continuum hypothesis corollary countable decision method defining axiom disjunction element elementarily equivalent elementary extension equality axioms equality theorem equivalent explicit definition expression numbers extension by definitions F is defined F is recursive finitary follows function F function or predicate function symbol functions and predicates H-index Hence Higman homomorphism hyperarithmetical identity axioms implies induction hypothesis infinite interpretation introduce isomorphism J-forces Lemma M-set n-ary function n-type natural numbers negation nonempty nonlogical axioms nonlogical symbols obtain ordinal partial functional predicate symbol prenex form problem provable prove quantifiers R-formula recursive function recursive partial functional recursively enumerable set relation replacing result special constants strongly undecidable structure subgroup subset suppose tautological consequence tautology theorem theory transfinite induction unary predicate valid variable-free variables Vy(y