Book contents
- Frontmatter
- Contents
- Preface
- Thanks
- 1 What Gödel's Theorems say
- 2 Functions and enumerations
- 3 Effective computability
- 4 Effectively axiomatized theories
- 5 Capturing numerical properties
- 6 The truths of arithmetic
- 7 Sufficiently strong arithmetics
- 8 Interlude: Taking stock
- 9 Induction
- 10 Two formalized arithmetics
- 11 What Q can prove
- 12 IΔ0, an arithmetic with induction
- 13 First-order Peano Arithmetic
- 14 Primitive recursive functions
- 15 LA can express every p.r. function
- 16 Capturing functions
- 17 Q is p.r. adequate
- 18 Interlude: A very little about Principia
- 19 The arithmetization of syntax
- 20 Arithmetization in more detail
- 21 PA is incomplete
- 22 Gödel's First Theorem
- 23 Interlude: About the First Theorem
- 24 The Diagonalization Lemma
- 25 Rosser's proof
- 26 Broadening the scope
- 27 Tarski's Theorem
- 28 Speed-up
- 29 Second-order arithmetics
- 30 Interlude: Incompleteness and Isaacson's Thesis
- 31 Gödel's Second Theorem for PA
- 32 On the ‘unprovability of consistency’
- 33 Generalizing the Second Theorem
- 34 Löb's Theorem and other matters
- 35 Deriving the derivability conditions
- 36 ‘The best and most general version’
- 37 Interlude: The Second Theorem, Hilbert, minds and machines
- 38 μ-Recursive functions
- 39 Q is recursively adequate
- 40 Undecidability and incompleteness
- 41 Turing machines
- 42 Turing machines and recursiveness
- 43 Halting and incompleteness
- 44 The Church–Turing Thesis
- 45 Proving the Thesis?
- 46 Looking back
- Further reading
- Bibliography
- Index
46 - Looking back
- Frontmatter
- Contents
- Preface
- Thanks
- 1 What Gödel's Theorems say
- 2 Functions and enumerations
- 3 Effective computability
- 4 Effectively axiomatized theories
- 5 Capturing numerical properties
- 6 The truths of arithmetic
- 7 Sufficiently strong arithmetics
- 8 Interlude: Taking stock
- 9 Induction
- 10 Two formalized arithmetics
- 11 What Q can prove
- 12 IΔ0, an arithmetic with induction
- 13 First-order Peano Arithmetic
- 14 Primitive recursive functions
- 15 LA can express every p.r. function
- 16 Capturing functions
- 17 Q is p.r. adequate
- 18 Interlude: A very little about Principia
- 19 The arithmetization of syntax
- 20 Arithmetization in more detail
- 21 PA is incomplete
- 22 Gödel's First Theorem
- 23 Interlude: About the First Theorem
- 24 The Diagonalization Lemma
- 25 Rosser's proof
- 26 Broadening the scope
- 27 Tarski's Theorem
- 28 Speed-up
- 29 Second-order arithmetics
- 30 Interlude: Incompleteness and Isaacson's Thesis
- 31 Gödel's Second Theorem for PA
- 32 On the ‘unprovability of consistency’
- 33 Generalizing the Second Theorem
- 34 Löb's Theorem and other matters
- 35 Deriving the derivability conditions
- 36 ‘The best and most general version’
- 37 Interlude: The Second Theorem, Hilbert, minds and machines
- 38 μ-Recursive functions
- 39 Q is recursively adequate
- 40 Undecidability and incompleteness
- 41 Turing machines
- 42 Turing machines and recursiveness
- 43 Halting and incompleteness
- 44 The Church–Turing Thesis
- 45 Proving the Thesis?
- 46 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 four main parts:
(a) The first part (Chapters 1 to 8), after explaining various key concepts, proves two surprisingly easy incompleteness theorems. Theorem 6.3 tells us that if T is a sound effectively axiomatized theory whose language is sufficiently expressive, then T can't be negation-complete. And Theorem 7.2 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 effectively 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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- An Introduction to Gödel's Theorems , pp. 367 - 369Publisher: Cambridge University PressPrint publication year: 2013