This book gathers together a colorful set of problems on classical Mathematical Logic, selected from over 30 years of teaching. The initial chapters start with problems from supporting fields, like set theory (ultrafilter constructions), full-information game theory (strategies), automata, and recursion theory (decidability, Kleene’s theorems). The work then advances toward propositional logic (compactness and completeness, resolution method), followed by first-order logic, including quantifier elimination and the Ehrenfeucht– Fraïssé game; ultraproducts; and examples for axiomatizability and non-axiomatizability. The Arithmetic part covers Robinson’s theory, Peano’s axiom system, and Gödel’s incompleteness theorems. Finally, the book touches universal graphs, tournaments, and the zero-one law in Mathematical Logic.
Instructors teaching Mathematical Logic, as well as students who want to understand its concepts and methods, can greatly benefit from this work. The style and topics have been specially chosen so that readers interested in the mathematical content and methodology could follow the problems and prove the main theorems themselves, including Gödel’s famous completeness and incompleteness theorems. Examples of applications on axiomatizability and decidability of numerous mathematical theories enrich this volume.
Author(s): Laszlo Csirmaz, Zalán Gyenis
Series: Problem Books in Mathematics
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 327
Tags: Logic; Logic Exercises
PREFACE
CONTENTS
Special Set Systems
Basic Constructions
Counterexamples
Set Systems of Functions
Filters
Ultrafilters
Games and Voting
Games
Voting
Formal Languages and Automata
Regular Languages and Automata
When the Context Does not Matter
Recursion Theory
Primitive Recursive Functions
Recursive Functions
Partial Recursive Functions
Coding
Universal Function
Decidability
Recursive Orders
Propositional Calculus
Formulas
Derivation
Coding
First-Order Logic
Basics
Expressing Properties
Models and Cardinalities
Ordered Sets
Coding
Fundamental Theorems
First-Order Derivations
Compactness and Other Properties
Elementary Equivalence
Basics
Ehrenfeucht–Fraïssé Game
Quantifier Elimination
Examples
Ultraproducts
What Ultraproducts Look Like
Applications
Advanced Exercises
Axiomatizability
Arithmetic
Robinson's Axiom System
Undecidability
Derivability
Peano's Axiom System
Arithmetical Hierarchy
Selected Applications
Independent Unary Relations
Universal Graphs
Universal Tournaments
Zero-One Law
Solutions
Special Set Systems
Games and Voting
Formal Languages and Automata
Recursion Theory
Propositional Calculus
First-Order Logic
Fundamental Theorems
Elementary Equivalence
Ultraproducts
Arithmetic
Selected Applications
INDEX
