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
16 - PA is incomplete
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
The pieces we need are finally all in place. So in this chapter we at long last learn how to construct ‘Gödel sentences’ and use them to prove that PA is incomplete. Then in the next chapter, we show how our arguments can be generalized to prove that PA – and any other formal arithmetic satisfying very modest constraints – is not only incomplete but incompletable. Our initial discussion in these two chapters uses ideas from Gödel's own treatment in 1931. Then in Chapter 19, we start extending Gödel's work.
The beautiful proofs now come thick and fast: savour them slowly!
Reminders
We start with some quick reminders – or bits of headline news, if you have impatiently been skipping forward in order to get to the exciting stuff.
Fix on some acceptable scheme for coding up wffs of PA by using Gödel numbers (‘g.n.’), and coding up sequences of wffs by super Gödel numbers. Then,
i. The diagonalization of ϕ is ∃y(y = ⌜ϕdlcorn ∧ ϕ), where ‘⌜ϕdlcorn’ here stands in for the numeral for ϕ's g.n. – the diagonalization of ϕ(y) is thus equivalent to ϕ(⌜ϕdlcorn). (Section 15.5)
ii. diag(n) is a p.r. function which, when applied to a number n which is the g.n. of some wff ϕ, yields the g.n. of ϕ's diagonalization. (Section 15.6)
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- Information
- An Introduction to Gödel's Theorems , pp. 138 - 146Publisher: Cambridge University PressPrint publication year: 2007