
Computability
Computable Functions, Logic, and the Foundations of Mathematics
- 382 pages
- English
- PDF
- Available on iOS & Android
Computability
Computable Functions, Logic, and the Foundations of Mathematics
About this book
Now in a new edition!--the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of Gödel's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics".
Frequently asked questions
- Essential is ideal for learners and professionals who enjoy exploring a wide range of subjects. Access the Essential Library with 800,000+ trusted titles and best-sellers across business, personal growth, and the humanities. Includes unlimited reading time and Standard Read Aloud voice.
- Complete: Perfect for advanced learners and researchers needing full, unrestricted access. Unlock 1.4M+ books across hundreds of subjects, including academic and specialized titles. The Complete Plan also includes advanced features like Premium Read Aloud and Research Assistant.
Please note we cannot support devices running on iOS 13 and Android 7 or earlier. Learn more about using the app.
Information
Table of contents
- Table of Contents
- Preface
- 1 Paradoxes
- 2 What Do the Paradoxes Mean?
- 3 Whole Numbers
- 4 Functions
- 5 Proofs
- 6 Infinite Collections?
- 7 Hilbert "On the Infinite"
- 8 Computability
- 9 Turing Machines
- 10 The Most Amazing Fact and Church's Thesis
- 11 Primitive Recursive Functions
- 12 The Grzegorczyk Hierarchy
- 13 Multiple Recursion
- 14 The Least Search Operator
- 15 Partial Recursive Functions
- 16 Numbering the Partial Recursive Functions
- 17 Listability
- 18 Turing Machine Computable = Partial Recursive
- 19 Propositional Logic
- 20 An Overview of First-Order Logic and Gödel's Theorems
- 21 First-Order Arithmetic
- 22 Functions Representable in Formal Arithmetic
- 23 The Undecidability of Arithmetic
- 24 The Unprovability of Consistency
- 25 Church's Thesis
- 26 Constructivist Views of Mathematics
- 27 Mathematics as Modeling
- Computability and Undecidability—A Timeline
- Bibliography
- Glossary and Index of Notation
- Index