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
38 - μ-Recursive functions
- 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
This chapter introduces the notion of a μ-recursive function – which is a natural extension of the idea of a primitive recursive function. Plausibly, the effectively computable functions are exactly the μ-recursive functions (and likewise, the effectively decidable properties are exactly those with μ-recursive characteristic functions).
Minimization and μ-recursive functions
The primitive recursive functions are the functions which can be defined using composition and primitive recursion, starting from the successor, zero, and identity functions. These functions are computable. But they are not the only computable functions defined over the natural numbers (see Section 14.5 for the neat diagonal argument which proves this). So the natural question to ask is: what other ways of defining new functions from old can we throw into the mix in order to get a broader class of computable numerical functions (hopefully, to get all of them)?
As explained in Section 14.4, p.r. functions can be calculated using bounded loops (as we enter each ‘for’ loop, we state in advance how many iterations are required). But as Section 4.6 illustrates, we also count unbounded search procedures – implemented by ‘do until’ loops – as computational. So, the obvious first way of extending the class of p.r. functions is to allow functions to be defined by means of some sort of ‘do until’ procedure. We'll explain how to do this in four steps.
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- Information
- An Introduction to Gödel's Theorems , pp. 285 - 296Publisher: Cambridge University PressPrint publication year: 2013