- •Contents
- •Preface to the Fifth Edition
- •1 Enumerability
- •1.1 Enumerability
- •1.2 Enumerable Sets
- •Problems
- •2 Diagonalization
- •Problems
- •3 Turing Computability
- •Problems
- •4 Uncomputability
- •4.1 The Halting Problem
- •4.2* The Productivity Function
- •Problems
- •5 Abacus Computability
- •5.1 Abacus Machines
- •5.2 Simulating Abacus Machines by Turing Machines
- •5.3 The Scope of Abacus Computability
- •Problems
- •6 Recursive Functions
- •6.1 Primitive Recursive Functions
- •6.2 Minimization
- •Problems
- •7 Recursive Sets and Relations
- •7.1 Recursive Relations
- •7.2 Semirecursive Relations
- •7.3* Further Examples
- •Problems
- •8.1 Coding Turing Computations
- •8.2 Universal Turing Machines
- •8.3∗ Recursively Enumerable Sets
- •Problems
- •9.1 First-Order Logic
- •9.2 Syntax
- •Problems
- •10.1 Semantics
- •10.2 Metalogical Notions
- •Problems
- •11 The Undecidability of First-Order Logic
- •11.1 Logic and Turing Machines
- •11.2 Logic and Primitive Recursive Functions
- •11.3 Lemma
- •Problems
- •12 Models
- •12.1 The Size and Number of Models
- •12.2 Equivalence Relations
- •Problems
- •13 The Existence of Models
- •13.1 Outline of the Proof
- •13.2 The First Stage of the Proof
- •13.3 The Second Stage of the Proof
- •13.4 The Third Stage of the Proof
- •13.5* Nonenumerable Languages
- •Problems
- •14 Proofs and Completeness
- •14.1 Sequent Calculus
- •14.2 Soundness and Completeness
- •14.3* Other Proof Procedures and Hilbert’s Thesis
- •Problems
- •15 Arithmetization
- •15.1 Arithmetization of Syntax
- •Problems
- •16 Representability of Recursive Functions
- •16.2 Minimal Arithmetic and Representability
- •16.3 Mathematical Induction
- •16.4* Robinson Arithmetic
- •Problems
- •17.1 The Diagonal Lemma and the Limitative Theorems
- •17.2 Undecidable Sentences
- •17.3* Undecidable Sentences without the Diagonal Lemma
- •Problems
- •18 The Unprovability of Consistency
- •Historical Remarks
- •19 Normal Forms
- •19.1 Disjunctive and Prenex Normal Forms
- •19.2 Skolem Normal Form
- •19.3 Herbrand’s Theorem
- •19.4 Eliminating Function Symbols and Identity
- •Problems
- •20 The Craig Interpolation Theorem
- •20.1 Craig’s Theorem and Its Proof
- •20.2 Robinson’s Joint Consistency Theorem
- •20.3 Beth’s Definability Theorem
- •Problems
- •21 Monadic and Dyadic Logic
- •21.1 Solvable and Unsolvable Decision Problems
- •21.2 Monadic Logic
- •21.3 Dyadic Logic
- •Problems
- •22 Second-Order Logic
- •Problems
- •23.2 Arithmetical Definability and Forcing
- •Problems
- •24 Decidability of Arithmetic without Multiplication
- •Problems
- •25 Nonstandard Models
- •25.1 Order in Nonstandard Models
- •25.2 Operations in Nonstandard Models
- •25.3 Nonstandard Models of Analysis
- •Problems
- •26 Ramsey’s Theorem
- •Problems
- •27 Modal Logic and Provability
- •27.1 Modal Logic
- •27.2 The Logic of Provability
- •27.3 The Fixed Point and Normal Form Theorems
- •Problems
- •Annotated Bibliography
- •General Reference Works
- •Textbooks and Monographs
- •By the Authors
- •Index
340 |
MODAL LOGIC AND PROVABILITY |
GL for all the following sentences D, of which the first is A and the last H:
B( C1( p), C2( p), . . . , Cn+1( p))
B( C1(H1), C2( p), . . . , Cn+1( p))
B( C1(H1), C2(H2), . . . , Cn+1( p))
.
.
.
B( C1(H1), C2(H2), . . . , Cn+1(Hn+1)).
Thus ( p ↔ A) → ( p ↔ H ) is a theorem of GL, to complete the proof of the fixed point theorem.
The normal form and fixed point theorems are only two of the many results about GL and related systems that have been obtained in the branch of logical studies known as provability logic.
Problems
27.1Prove the cases of Theorem 27.1 that were ‘left to the reader’.
27.2Let S5 = K + (A1) + (A2) + (A3). Introduce an alternative notion of model for S5 in which a model is just a pair W = (W, ω) and W, w |= A iff W, v |= A for all v in W . Show that S5 is sound and complete for this notion of model.
27.3Show that in S5 every formula is provably equivalent to one such that in a subformula of form A, there are no occurrences of in A.
27.4Show that there is an infinite transitive, irreflexive model in which the sentence( p → p) → p is not valid.
27.5Verify the entries in Table 27-1.
27.6Suppose for A in Table 27-1 we took ( p → ) → ( p → ). What would be the corresponding H ?
27.7To prove that the Godel¨ sentence is not provable in P, we have to assume the consistency of P. To prove that the negation of the Godel¨ sentence is not provable in P, we assumed in Chapter 17 the ω-consistency of P. This is a stronger assumption than is really needed for the proof. According to Table 27-1, what assumption is just strong enough?