Contents -- THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS -- THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS, II* -- MARGINALIA TO A THEOREM OF SILVER -- 0 Introduction -- 1 The approach -- 2 Fine structure lemmas -- 3 The non w cofinal case -- 4 Vicious sequences -- 5 The w cofinal case -- Bibliography -- THREE THEOREMS ON RECURSIVE ENUMERATION. I. DECOMPOSITION. II. MAXIMAL SET. III. ENUMERATION WITHOUT DUPLICATION -- BIBLIOGRAPHY -- HIGHER SET THEORY AND MATHEMATICAL PRACTICE * -- Introduction -- Section 1 -- Section 2 -- Section 3 -- Section 4 -- References -- INTRODUCTION TO -LOGIC -- 1. SYSTEMS OF DENOTATIONS -- 2. DILATORS -- 3. THE ALGEBRAIC THEORY OF DILATORS -- 4. DILATORS AS WELL-ORDERED CLASSES -- 5. A AND TRADITIONAL PROOF-THEORY -- 6. ß-PROOFS -- 7. INDUCTIVE LOGIC -- 8. APPLICATIONS TO GENERALIZED RECURSION -- 9. DESCRIPTIVE SET-THEORY -- BIBLIOGRAPHY -- 1. General Expositions -- 2. Relations to Generalized Recursion -- 3. Dilators and Related Concepts -- 4. Cut-elimination -- 5. A and Related Topics -- CONSISTENCY-PROOF FOR THE GENERALIZED CONTINUUM-HYPOTHESIS¹ -- THE MORDELL-LANG CONJECTURE FOR FUNCTION FIELDS -- 1. INTRODUCTION -- 2. THIN TYPES AND ZARISKI GEOMETRIES -- 3. MANIN'S HOMOMORPHISM AND BUIUM'S REDUCTION -- 4. ABELIAN GROUPS OF FINITE MORLEY DIMENSION -- 5. THE MAIN THEOREM -- 6. A QUESTION OF VOLOCH'S -- REFERENCES -- MODEL-THEORETIC INVARIANTS: APPLICATIONS TO RECURSIVE AND HYPERARITHMETIC OPERATIONS -- Introduction. -- 1. Basic Notions. -- 2. Invariants. -- 2.1. Technical lemmas on general models. -- 2.2. Technical lemmas on w-models. -- 3. Applications. -- 3.1. Theory of w-models. -- 3.2. Recursion theory on the recursive ordinals. -- RECURSIVE FUNCTIONALS AND QUANTIFIERS OF FINITE TYPES I -- 1. Primitive recursive functions. -- 2. Alterations of quantifiers. -- 3. Partial and general recursive functions.

4. Construction of indices. -- 5. Reduction of the inductive definition of {z} (a)~w to an explicit definition. -- 6. Reduction in type of a quantifier. -- 7. Predicates of order r. -- 8. µ-recursiveness versus general recursiveness. -- BIBLIOGRAPHY -- A RECURSIVELY ENUMERABLE DEGREE WHICH WILL NOT SPLIT OVER ALL LESSER ONES -- 0. Introduction -- 1. Formulation of the problem -- 2. The priority tree and the construction -- 3. Properties of the construction -- 4. Verification of Propositions 1 to 10 -- References -- ENUMERABLE SETS ARE DIOPHANTINE -- ADDENDUM * -- BIBLIOGRAPHY -- CATEGORICITY IN POWER(¹) -- Introduction. -- 1. Preliminaries. -- 2. Transcendence in rank. -- 3. Results depending on Ramsey's theorem. -- 4. Models of totally transcendental theories. -- 5. Saturated models and categoricity in power. -- REFERENCES -- HYPERANALYTIC PREDICATES -- 1. A hyperanalytic function not recursive in any H 2 (a). -- 2. Functions recursive in a type-3 object. -- 3. The hyperanalytic hierarchy. -- 4. Minimum functions. -- 5. Predicates r.e. in ³E. -- 6. Normal forms for predicates r.e. in ³E. -- 7. An extension of the hyperanalytic hierarchy. -- 8. Comments on results for types other than 3. -- BIBLIOGRAPHY -- SOLUTION OF POST'S REDUCTION PROBLEM AND SOME OTHER PROBLEMS OF THE THEORY OF ALGORITHMS. I.¹) -- CONTENTS -- Introduction -- CHAPTER I. Functional Representation of Partial Recursive Operators -- 1. Sequences and Quasisequences -- 2. Functional Representation of Operators -- 3. A Universal Partial Recursive Operator -- 4. The Calculus of M-problems -- CHAPTER II. Decision Problems of Recursively Enumerable Sets -- 1. The Semilattice U(P) -- 2. Post's Reduction Problem -- BIBLIOGRAPHY -- RECURSIVELY ENUMERABLE SETS OF POSITIVE INTEGERS AND THEIR DECISION PROBLEMS -- Introduction. -- 1. Recursive versus recursively enumerable sets.

2. A form of Godel's theorem. -- 3. The complete set K -- creative sets. -- 4. One-one reductibility, to K -- many-one reducibility. -- 5. Simple sets. -- 6. Reducibility by truth-tables. -- 7. Non-reducibility of creative sets to simple sets by bounded truth-tables. -- 8. Counter-example for unbounded truth-tables. -- 9. Hyper-simple sets. -- 10. Non-reducibility of creative sets to hyper-simple sets by truth-tables unrestricted. -- 11. General (Turing) reducibility. -- BIBLIOGRAPHY -- NON-STANDARD ANALYSIS -- 1. Introduction. -- 2. Non-standard analysis and non-archimedean fields. -- 3. Examples in non-standard Analysis. -- REFERENCES -- STABLE THEORIES -- 0. Introduction -- 1. Notations. -- 2. On possible cardinalities of S(A) -- 3. On some properties of stable theories. -- 4. On categorical elementary and pseudo elementary classes. -- REFERENCES -- THE PROBLEM OF PREDICATIVITY -- REFERENCES -- ON DEGREES OF RECURSIVE UNSOLVABILITY* -- 1. Relations involving both join and jump operations -- 2. Sets of degrees without g.l.b. or without l.u.b. -- 3. Non-density -- REFERENCES -- A DECISION METHOD FOR ELEMENTARY ALGEBRA AND GEOMETRY -- INTRODUCTION -- SECTION 1. THE SYSTEM OF ELEMENTARY ALGEBRA -- SECTION 2. DECISION METHOD FOR ELEMENTARY ALGEBRA -- SECTION 3. EXTENSIONS TO RELATED SYSTEMS -- NOTES -- BIBLIOGRAPHY -- SUPPLEMENTARY NOTES -- MODEL COMPLETENESS RESULTS FOR EXPANSIONS OF THE ORDERED FIELD OF REAL NUMBERS BY RESTRICTED PFAFFIAN FUNCTIONS AND THE EXPONENTIAL FUNCTION -- 1. INTRODUCTION -- First Main Theorem. -- Second Main Theorem. -- 2. TOWARDS THE PROOF OF THE FIRST MAIN THEOREM -- 3. RESULTS OF KHOVANSKII AND VAN DEN DRIES -- 4. DlFFERENTIABLE GERMS IN ARBITRARY EXPANSIONS OF R -- 5. DEFINABLE POINTS ON COMPONENTS AND THE PROOF OF LEMMA 2.7 -- 6. ONE DIMENSIONAL VARIETIES -- 7. THE PROOF OF LEMMA 2.8 -- 8. THE PROOF OF LEMMA 2.9.

9. TOWARDS THE PROOF OF THE SECOND MAIN THEOREM -- 10. SMOOTH 0-MINIMAL THEORIES -- 11. BOUNDING THE SOLUTIONS TO EXPONENTIAL-POLYNOMIAL EQUATIONS AND THE COMPLETION OF THE PROOF OF THE SECOND MAIN THEOREM -- REFERENCES -- STRUCTURAL PROPERTIES OF MODELS OF N1-CATEGORICAL THEORIES -- 1. -- 2. Proofs -- References -- Permissions.