- •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
PROBLEMS |
317 |
(In general, there could be more than two as for each U , but the proof would be no different.) Now let
H (x, u1, u2, v11, v12, v21, v22)
be the result of replacing any subformula of form Ui (w) by Gi (w, vi1, vi2). Then
T = {b: M |= H [b, s1, s2, a11, a12, a21, a22]}
and is parametrically definable as required.
Problems
25.1Show how the proof of the existence of averages can be formalized in P, in the style of Chapter 16.
25.2Show that there is a recursive relation on the natural numbers that is also isomorphic to the order <K on the set K defined in the proof of Theorem 25.1.
25.3Show that the successor function † associated with may also be taken to be recursive.
25.4Show that in an -model that is not an ω-model, the upper domain cannot contain all subsets of the lower domain.
The remaining problems outline the proof of the arithmetical Lowenheim¨– Skolem theorem, and refer to the alternative proof of the model existence lemma in section 13.5 and the problems following it.
25.5Assuming Church’s thesis, explain why, if is a recursive set of (code numbers of) sentences in a recursive language, the set * obtained by adding (the code numbers of) the Henkin sentences to is still recursive (assuming a suitable coding of the language with the Henkin constants added).
25.6Explain why, if is an arithmetical set of sentences, then the relation
i codes a finite set of sentences , j codes a sentence D,
and implies D
is also arithmetical.
25.7Suppose * is a set of sentences in a language L* and i0, i1, . . . an enumeration of all the sentences of L*, and suppose we form # as the union of sets n ,
where 0 = * and n+1 = n if n implies in , while n+1 = n {in } otherwise. Explain why, if * is arithmetical, then # is arithmetical.
25.8Suppose we have a language with relation symbols and enumerably many constants c0, c1, . . . , but function symbols and identity are absent. Suppose
# is arithmetical and has the closure properties required for the construction of section 13.2. In that construction take as the element ciM associated with the constant ci the number i. Explain why the relation RM associated with any relation symbol R will then be arithmetical.
25.9Suppose we have a language with relation symbols and enumerably many
constants c0, c1, . . . , but that function symbols are absent, though identity may be present. Suppose # is arithmetical has the closure properties required
318 |
NONSTANDARD MODELS |
for the construction of section 13.3. Call i minimal if there is no j < i such that ci = c j is in #. Show that the function δ taking n to the nth number i such that ci is minimal is arithmetical.
25.10Continuing the preceding problem, explain why for every constant c there is a unique n such that c = δ(n) is in #, and that if in the construction of
section 13.3 we take as the element ciM associated with the constant ci this number n, then the relation RM associated with any relation symbol R will be arithmetical.
25.11Explain how the arithmetical Lowenheim¨ –Skolem theorem for the case where function symbols are absent follows on putting together the preceding six problems, and indicate how to extend the theorem to the case where they are present.