Computability
Computable Functions, Logic, and the Foundations of Mathematics
3rd Edition
Published on 24. October 2018
382 pages
978-0-9815507-3-2 (ISBN)
Description
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".
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Richard L. Epstein | Walter A. Carnielli
Computability
Computable Functions, Logic, and the Foundations of Mathematics
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06/2008
Advanced Reasoning Forum
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Persons
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.
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.
Content
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 Undecidability-A Timeline
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 Undecidability-A Timeline
