Book:Yu.I. Manin/A Course in Mathematical Logic

Yu.I. Manin: A Course in Mathematical Logic
Published 1977, Springer Verlag, ISBN 0-387-90243-0.

Subject Matter

 * Mathematical Logic

Contents

 * Part I: PROVABILITY
 * I Introduction to formal languages
 * 1 General information
 * 2 First order languages
 * Digression: names
 * 3 Beginner's course in translation
 * Digression: syntax


 * II Truth and deducibility
 * 1 Unique reading lemma
 * 2 Interpretation: truth, definability
 * 3 Syntactic properties of truth
 * Digression: natural logic
 * 4 Deducibility
 * Digression: proof
 * 5 Tautologies and Boolean algebras
 * Digression: Kennings
 * 6 Gödel's completeness theorem
 * 7 Countable models and Skolem's paradox
 * 8 Language extensions
 * 9 Undefinability of truth: the language SELF
 * 10 Smullyan's language of arithmetic
 * 11 Undefinability of truth: Tarski's theorem
 * Digression: self-reference
 * 12 Quantum logic
 * Appendix: The von Neumann Universe


 * III The continuum problem and forcing
 * 1 The problem: results, ideas
 * 2 A language of real analysis
 * 3 The continuum hypothesis is not deducible in $\mathrm{L_2Real}$
 * 4 Boolean-valued universes
 * 5 The axiom of extensionality is "true"
 * 6 The axioms of pairing, union, power set, and regularity are "true"
 * 7 The axioms of infinity, replacement, and choice are "true"
 * 8 The continuum hypothesis is "false" for suitable $B$
 * 9 Forcing


 * IV The continuum problem and constructible sets
 * 1 Gödel's constructible universe
 * 2 Definability and absoluteness
 * 3 The constructible universe as a model for set theory
 * 4 The generalized continuum hypothesis is $L$-true
 * 5 Constructibility formula
 * 6 Remarks on formalization
 * 7 What is the cardinality of the continuum?


 * Part II: COMPUTABILITY
 * V Recursive functions and Church's thesis
 * 1 Introduction. Intuitive computability
 * 2 Partial recursive functions
 * 3 Basic examples of recursiveness
 * 4 Enumerable and decidable sets
 * 5 Elements of recursive geometry


 * VI Diophantine sets and algorithmic undecidability
 * 1 The basic result
 * 2 Plan of proof
 * 3 Enumerable sets are $D$-sets
 * 4 The reduction
 * 5 Construction of a special Diophantine set
 * 6 The graph of the exponential is Diophantine
 * 7 The graphs of the functorial and the binomial coefficients are Diophantine
 * 8 Versal families
 * 9 Kolmogorov complexity


 * Part III: PROVABILITY AND COMPUTABILITY
 * VII Gödel's incompleteness theorem
 * 1 Arithmetic of syntax
 * 2 Incompleteness principles
 * 3 Nonenumerability of true formulas
 * 4 Syntactic analysis
 * 5 Enumerability of deducible formulas
 * 6 The arithmetical hierarchy
 * 7 Productivity of arithmetical truth
 * 8 On the length of proofs


 * VIII Recursive groups
 * 1 Basic result and its corollaries
 * 2 Free products and HNN-extensions
 * 3 Embeddings in groups with two generators
 * 4 Benign subgroups
 * 5 Bounded systems of generators
 * 6 End of the proof


 * Index