Introduction to Model TheoryCRC Press, 31 oct 2000 - 324 páginas Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory. |
Índice
32 | 3 |
Languages | 15 |
Semantics | 21 |
Beginnings of model theory | 39 |
First consequences of the finiteness theorem | 49 |
Malcevs applications to group theory | 67 |
Some theory of orderings | 87 |
Basic properties of theories | 109 |
Theories and types | 161 |
Thick and thin models | 185 |
Countable complete theories | 195 |
Two applications | 205 |
15 | 239 |
Hints to selected exercises | 269 |
Solutions for selected exercises | 277 |
293 | |
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Términos y frases comunes
1-types abelian group algebraically closed fields algebraically compact arbitrary atomic formulas automorphism axiom axiomatized bijection boolean algebra Chapter complete theory consider constants contains Corollary countable countable model deductive closure definable denote dimension division ring element elementarily equivalent elementarily prime model elementary extension elementary substructure embedding Exercise filter finite models finite subset finiteness theorem free variables given hence homomorphism implies induction isolated isomorphic L-formula L-sentences L-structure L-theory language Lemma Lindenbaum-Tarski algebra linear ordering Löwenheim-Skolem Math model theory models of TZ n-tuple natural number nonempty notation parameters partial ordering polynomial prime model Proof Proposition Prove quantifier elimination quantifier-free realized Remark satisfying saturated model set of sentences signature strongly minimal strongly minimal theory structure subgroup Th(M torsionfree totally categorical trivial tuple tuple ā ultrafilter uncountable vector spaces well-ordering yields
Referencias a este libro
A Concise Introduction to Mathematical Logic Wolfgang Rautenberg No hay ninguna vista previa disponible - 2006 |
Approximations and Endomorphism Algebras of Modules Rüdiger Göbel,Jan Trlifaj No hay ninguna vista previa disponible - 2006 |