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".

Richard L. Epstein received his Ph.D. in mathematics at the University of California, Berkeley. He was a postdoctoral fellow in mathematics and philosophy at Victoria University of Wellington, New Zealand, a U.S. National Academy of Sciences Scholar to Poland, a Fulbright Fellow to Brazil, and a CNPQ Fellow to the University of Paraiba, Brazil. He is currently the Head of the Advanced Reasoning Forum. Walter A Carnielli received his Ph.D in logic and the foundations of mathematics at the State University of Campinas, Brazil. He has held a postdoctoral fellowship at the University of California, Berkeley, and an Alexander von Humboldt scholar to Universitat Bonn. From 1999 to 2017 he was Director of the Center for Logic, Epistemology, and the History of Science at the State University of Campinas, Brazil.

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 Theorem 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 UndecidabilityA Timeline