- •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
294 |
ARITHMETICAL DEFINABILITY |
A-correct p forces S. By Lemma 23.6, there exists a (fully) generic set A* such that p is A*-correct. Since p forces S, by Lemma 23.7 (in its original version), NAG |= S. But this means A* is arithmetical, contrary to Lemma 23.8.
Problems
23.1Use Beth’s definability theorem, Tarski’s theorem on the first-order indefinability of first-order arithmetic truth, and the results of section 23.1 to obtain another proof of the existence of nonstandard models of arithmetic.
23.2Show that for each n the set of (code numbers of) true prenex sentences of the language of arithmetic that contain at most n quantifiers is arithmetical. Show the same with ‘prenex’ omitted.
23.3Show that if p B, then p B.
23.4Given an example of a sentence B such that the set of even numbers FORCES neither B nor B.
23.5Show that the set of pairs (i, j) such that j codes a sentence of LG and i codes a condition that forces that sentence is not arithmetical.
23.6Where would the proof of Addison’s theorem have broken down if we had worked with , & , rather than , , (and made the obvious analogous stipulations in the definition of forcing)?
23.7Show that the only arithmetical subsets of a generic set are its finite subsets.
23.8Show that if A is generic, then { A} is not arithmetical.
23.9Show that { A : A is generic} is not arithmetical.
23.10Show that every generic set contains infinitely many prime numbers.
23.11Show that the class of generic sets is nonenumerable.
23.12A set of natural numbers is said to have density r, where r is a real number, if r is the limit as n goes to infinity of the ratio (number of members of A < n)/n. Show that no generic set has a density.