Book contents
- Frontmatter
- Contents
- Preface
- 1 What Gödel's Theorems say
- 2 Decidability and enumerability
- 3 Axiomatized formal theories
- 4 Capturing numerical properties
- 5 The truths of arithmetic
- 6 Sufficiently strong arithmetics
- 7 Interlude: Taking stock
- 8 Two formalized arithmetics
- 9 What Q can prove
- 10 First-order Peano Arithmetic
- 11 Primitive recursive functions
- 12 Capturing p.r. functions
- 13 Q is p.r. adequate
- 14 Interlude: A very little about Principia
- 15 The arithmetization of syntax
- 16 PA is incomplete
- 17 Gödel's First Theorem
- 18 Interlude: About the First Theorem
- 19 Strengthening the First Theorem
- 20 The Diagonalization Lemma
- 21 Using the Diagonalization Lemma
- 22 Second-order arithmetics
- 23 Interlude: Incompleteness and Isaacson's conjecture
- 24 Gödel's Second Theorem for PA
- 25 The derivability conditions
- 26 Deriving the derivability conditions
- 27 Reflections
- 28 Interlude: About the Second Theorem
- 29 µ-Recursive functions
- 30 Undecidability and incompleteness
- 31 Turing machines
- 32 Turing machines and recursiveness
- 33 Halting problems
- 34 The Church–Turing Thesis
- 35 Proving the Thesis?
- 36 Looking back
- Further reading
- Bibliography
- Index
36 - Looking back
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 What Gödel's Theorems say
- 2 Decidability and enumerability
- 3 Axiomatized formal theories
- 4 Capturing numerical properties
- 5 The truths of arithmetic
- 6 Sufficiently strong arithmetics
- 7 Interlude: Taking stock
- 8 Two formalized arithmetics
- 9 What Q can prove
- 10 First-order Peano Arithmetic
- 11 Primitive recursive functions
- 12 Capturing p.r. functions
- 13 Q is p.r. adequate
- 14 Interlude: A very little about Principia
- 15 The arithmetization of syntax
- 16 PA is incomplete
- 17 Gödel's First Theorem
- 18 Interlude: About the First Theorem
- 19 Strengthening the First Theorem
- 20 The Diagonalization Lemma
- 21 Using the Diagonalization Lemma
- 22 Second-order arithmetics
- 23 Interlude: Incompleteness and Isaacson's conjecture
- 24 Gödel's Second Theorem for PA
- 25 The derivability conditions
- 26 Deriving the derivability conditions
- 27 Reflections
- 28 Interlude: About the Second Theorem
- 29 µ-Recursive functions
- 30 Undecidability and incompleteness
- 31 Turing machines
- 32 Turing machines and recursiveness
- 33 Halting problems
- 34 The Church–Turing Thesis
- 35 Proving the Thesis?
- 36 Looking back
- Further reading
- Bibliography
- Index
Summary
Let's finish by taking stock one last time. At the end of the last Interlude, we gave a road-map for the final part of the book. So we won't repeat the gist of that detailed local guide to recent chapters; instead, we'll stand further back and give a global overview. And let's concentrate on the relationship between our various proofs of incompleteness. Think of the book, then, as falling into three main parts:
(a) The first part (Chapters 1 to 7), after explaining various key concepts, proves two surprisingly easy incompleteness theorems. Theorem 5.7 tells us that if T is a sound axiomatized theory whose language is sufficiently expressive, then T can't be negation complete. And Theorem 6.2 tells us that we can weaken the soundness condition and require only consistency if we strengthen the other condition (from one about what T can express to one about what it can prove): if T is a consistent axiomatized theory which is sufficiently strong, then T again can't be negation complete.
Here the ideas of being sufficiently expressive/sufficiently strong are defined in terms of expressing/capturing enough effectively decidable numerical properties or relations. So the arguments for our two initial incompleteness theorems depend on a number of natural assumptions about the intuitive idea of effective decidability. And the interest of those theorems depends on the assumption that being sufficiently expressive/sufficiently strong is a plausible desideratum on formalized arithmetics.
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- Information
- An Introduction to Gödel's Theorems , pp. 342 - 343Publisher: Cambridge University PressPrint publication year: 2007