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
40 - Undecidability and incompleteness
- 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
With a bit of help from Church's Thesis, our Theorem 39.2 – Q is recursively adequate – very quickly yields two new Big Results: first, any nice theory is undecidable; and second, theoremhood in first-order logic is undecidable too.
The old theorem that Q is p.r. adequate is, of course, the key result which underlies our previous incompleteness theorems for theories that are p.r. axiomatized and extend Q. Our new theorem correspondingly underlies some easy (but unexciting) generalizations to recursively axiomatized theories that extend Q. More interestingly, we can now prove a formal counterpart to the informal incompleteness theorem of Chapter 7.
Some more definitions
We pause for some reminders, interlaced with definitions for some fairly self-explanatory bits of new jargon.
(a) First, recall from Section 3.2 the informal idea of a decidable property, i.e. a property P whose characteristic function cP is computable. In Section 14.6 we introduced a first formal counterpart to this idea, the notion of a p.r. property, i.e. one with a p.r. characteristic function. However, there can be decidable properties which aren't p.r. (as not all effective computations deliver primitive recursive functions). We now add:
A numerical property P is recursively decidable iff its characteristic function cP is μ-recursive.
That it is to say, P is recursively decidable iff there is a μ-recursive function which, given input n, delivers a 0/1, yes/no, verdict on whether n is P. The definition obviously extends in a natural way to cover recursively decidable numerical relations.
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- An Introduction to Gödel's Theorems , pp. 300 - 309Publisher: Cambridge University PressPrint publication year: 2013