Model Theory MARIA MANZANO. University of Salamanca, Spain. Translated by RUY J. G. B. DE QUEIROZ

Transcription

1 Model Theory MARIA MANZANO University of Salamanca, Spain Translated by RUY J. G. B. DE QUEIROZ CLARENDON PRESS OXFORD 1999

2 Contents Glossary of symbols and abbreviations General introduction 1 xix Basic notions: universal algebra Introduction Structures Some well-known structures Group Rings and fields Ring Field Order Chain, bound, supremum, and infimum Well-order Peano structures 1.2. Boolean algebras Atoms of a Boolean algebra Boolean rings Lattices Filters and ideals Relations between structures without using the formal language Substructure Reduct \ Homomorphism Embedding Isomorphism 1.3. Onto homomorphism Points to remember Summary First order languages: semantics Introduction First order language adequate to a structure Alphabet Terms and formulas Notational conventions ,

3 xvi Contents Omission of brackets 39 Other abbreviations Induction 41 Proofs by induction on the complexity of terms/ formulas 42 Definitions by recursion Bound and free occurrences Interpretation of a language in a structure Interpretation Consequence Validity Satisfiability Logical equivalence Substitution 48 Simultaneous substitution Extension by definition Some useful languages The language of identity The language of groups The language of orders The language of arithmetic The language of set theory (Zermelo-Fraenkel axioms) 1 Standard set hierarchy 1 Zermelo-Fraenkel axioms Semantic theorems The coincidence theorem The substitution theorem The isomorphism theorem Definability in a structure \ 70 3 Completeness of first order logic Introduction Deductive calculus Rules of the calculus Derived rules Syntactic notions Soundness of the deductive calculus Completeness theorem (countable languages) Scheme for the completeness theorem Henkin's theorem implies strong completeness Lindenbaum's lemma Henkin's lemma 91

4 Contents xvii Definition of structure Henkin's theorem Completeness of the calculus (L of arbitrary cardinality K) Conclusion Basic notions: model theory Introduction Elementary equivalence Elementary substructure Elementary embedding Theory Theory of a class of structures and models of a set of sentences Expansion by enumeration: diagrams The compactness theorem and its mathematical implications Introduction Axiomatizability Axiomatizable property Class of axiomatizable structures Axiomatizable theory Compactness (the method of diagrams) Some consequences of the compactness theorem Graphs , Elementary embeddings The construction of ultraproducts Direct product 152 Equivalence relation \ 153 Boolean models Reduced product The Los theorem and its corollaries Appendix: filters and ultrafilters 159 Lowenheim Skolem theorems and their consequences 13.0 Introduction 13.1 The structure of the chapter 15.2 Lowenheim-Skolem theorems 15.3 Nonstandard models Nonstandard models of Peano arithmetic 173 What a nonstandard model looks like from the inside 175

5 xviii Contents.3.2 Nonstandard models of the reals Construction of *R* Properties of 9t* What do the new elements of SH* look like? Properties of <K which are not expressible in L(9i) Other peculiarities of 9t* Finite numbers.4 Skolem's paradox.4.1 The mathematical universe.4.2 Axiomatic set theory The natural numbers in ZF set theory.4.3 The paradox of Skolem 7 Complete and categorical theories 7.0 Introduction 7.1 Completeness and categoricity 7.2 Quantifier elimination Theories which admit quantifier elimination Test for quantifier elimination 7.3 Model-completeness 7.4 The structure 9t s = (N, 0,s): completeness and decision procedure of its theory Models of A s Equivalence relation in a model of A s Quantifier elimination in T7i(9t s ) Appendix A: ordinals and cardinals A.I "- Ordinals A. 1.1 Transfinite induction A. 2 Cardinals \ A.2.1 Comparison of quantity A.2.2 Cardinal arithmetic A.2.3 Definition of the cardinals A.2.4 Finite and infinite cardinals A.2.5 Arithmetical properties of the cardinals Bibliography ' 227 Index 233