A Beginner's Further Guide to Mathematical Logic 9814730998, 9789814730990

Table of contents :
Contents
Preface
Part I: More on Propositional and First-Order Logic
Chapter 1. More on Propositional Logic
I. Propositional Logic and the Boolean Algebra of Sets
II. An Algebraic Approach
III. Another Completeness Proof
Consequence Relations
Consistency
Maximal Consistency
Boolean Consequence Relations
Compactness
IV. Fidelity to Modus Ponens
Solutions to the Problems of Chapter 1
Chapter 2. More on First-Order Logic
I. Magic Sets
Applications of Magic Sets
II. Gentzen Sequents and Some Variants
Background
Gentzen Type Axiom Systems
The System G0 in Uniform Notation
Completeness of G0
Block Tableau Inference Rules for Propositional Logic
A Gentzen Type System G1 for First-Order Logic
The System GG
The Completeness of GG
III. Craig's Lemma and an Application
Craig's Interpolation Lemma
Proof of Craig's Interpolation Lemma for the Gentzen System GG
Beth's Definability Theorem
IV. A Unification
V. A Henkin-Style Completeness Proof
Solutions to the Problems of Chapter 2
Part II: Recursion Theory and Metamathematics
Chapter 3. Some Special Topics
I. A Decision Machine
Universal Registers
Contra-Universal Registers
Creative Registers
Fixed Points
A Halting Problem
Domination
Affirmation Sets
II. Variations on a Theme of Godel
Provable, Refutable and Undecidable Sentences
Set Representation
Provability and Refutability Predicates
Creative Predicates
Fixed Points
Domination
III. R-Systems
Universal and Contra-Universal Sets
Creative R-Sets
Fixed Points
R-Separable R-Sets
Domination
IV. A Synthesis
The Synthesis System
Undecidable Numbers
Universal Numbers
Contra-Universal Numbers
Creative Numbers
Fixed Points
Domination
Set Representation
Solutions to the Problems of Chapter 3
Chapter 4. Elementary Formal Systems and Recursive Enumerability
I. More on Elementary Formal Systems
Representability
Some Basic Closure Properties
II. Recursive Enumerability
Finite Quantifications (Bounded Quantifiers)
A Recursive Pairing Function (x; y)
The Inverse Functions K and L
The n-tupling Functions n(x1, ... , xn)
The Inverse Functions Kni
III. A Universal System
The Recursive Unsolvability of (U)
Solutions to the Problems of Chapter 4
Chapter 5. Some Recursion Theory
I. Enumeration and Iteration Theorems
The Enumeration Theorem
Iteration Theorems
Maximal Indexing
II. Recursion Theorems
Unsolvable Problems and Rice's Theorem
Applications
The Strong Recursion Theorem
Creative, Productive, Generated and Universal Sets
Post's Sets C and K
Complete Productivity and Creativity
Generative Sets
Many-One Reducibility
Universal Sets
Uniformly Universal Sets
Weak Co-productivity
Discussion
Recursive Isomorphism
Solutions to the Problems of Chapter 5
Chapter 6. Doubling Up
Double Recursion Theorems
The Strong Double Recursion Theorem
Discussion
More Doubling Up
Eective Inseparability
Kleene's Construction
Doubly Generative Pairs
Doubly Generative, Doubly Universal
Double Co-productivity
Solutions to the Problems of Chapter 6
Chapter 7. Metamathematical Applications
I. Simple Systems
Intended Applications
Discussion
The Diagonal Function d(x)
Admissible Functions
Separation
Normal Systems
Fixed Points
II. Standard Simple Systems
Undecidability and Incompleteness
Recursive Inseparability
Rosser Systems
Effective Rosser Systems
Rosser Fixed-Point Property
Uniform Incompleteness
Solutions to the Problems of Chapter 7
Part III: Elements of Combinatory Logic
Chapter 8. Beginning Combinatory Logic
Fixed Points
The Composition Condition
Introducing the Combinator M, the Mocker
(the Mockingbird)
Egocentric Elements
Agreeable Elements
Compatible Elements
Smug Elements
Fixation
Narcissistic Elements
Introducing the Combinator K, the Kestrel
Introducing the Combinator L, the Lark
Discussion
Introducing the Combinator I,
the Identity Combinator
Sages
Solutions to the Problems of Chapter 8
Chapter 9. Combinatorics Galore
I. The B-Combinators
Introducing the B Combinator, the Bluebird
Some Derivatives of B
II. The Permuting Combinators
Introducing the Combinator T, the Thrush
Introducing the Combinator R, the Robin
Introducing the Combinator C, the Cardinal
Introducing the Combinator F, the Finch
ntroducing the Combinator V, the Vireo
Some Relations
Vireos Revisited
III. The Q-Family and the Goldfinch, G
Q1 and Q2
The Birds Q3 (Quirky), Q4 (Quacky), Q5 (Quintessential), Q6 (Quivering)
The Combinator G, the Goldfinch
IV. Combinators Derivable from B,T,M and I ( -I Combinators)
The Combinator W, the Warbler and the Combinator W0, the Converse Warbler
The Combinator H, the Hummingbird
Larks Revisited
The Combinator S, the Starling
The Order of a Combinator
The P Group of Combinators
The
Group of Combinators
Solutions to the Problems of Chapter 9
Chapter 10. Sages, Oracles and Doublets
Sages Again
The Turing Combinator
The Oracle O
Double Sages
The Nice Combinator N
Double Oracles
Solutions to the Problems of Chapter 10
Chapter 11. Complete and Partial Systems
I. The Complete System
Combinatorial Completeness
A Fixed Point Problem
Doubling Up
II. Partial Systems of Combinatory Logic
-I Combinators
B, T, I Combinators
Solutions to the Problems of Chapter 11
Chapter 12. Combinators, Recursion and the Undecidable
Logical Combinators
Arithmetic Combinators
The Zero Tester
I. Preparation for the Finale
Property Combinators
Relational Combinators
Functional Combinators
The Minimization Principle
The Length Measurer
Concatenation to the Base 10
II. The Grand Problem
Godel Numerals
Normalization
The Second Fixed Point Principle
Representability
Godel Sentences
Discussion
Inference Rules
Solutions to the Problems of Chapter 12
Afterword. Where to Go from Here
Rank
The Axiom of Choice and the Generalized Continuum Hypothesi
Constructible Sets
References
Index