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Achim Blumensath Blumens@Fi.Muni.Cz Is Document Achim Blumensath blumens@fi.muni.cz tis documentwas last updated --. te latest version can be found at www.fi.muni.cz/~blumens C Achim Blumensath tiswork is licensed under the Creative Commons Attri- bution . International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/./. Contents A. Setteory Basic set theory Sets and classes............................ Stages and histories ......................... te cumulative hierarchy...................... Relations Relations and functions....................... Products and unions ........................ Graphs and partial orders ..................... Fixed points and closure operators ................ Ordinals Well-orders.............................. Ordinals................................ Induction and fixed points..................... Ordinal arithmetic .......................... Zermelo-Fraenkelset theory te Axiom of Choice ........................ Cardinals............................... Cardinal arithmetic ......................... Cofinality ............................... , -- — © v Contents te Axiom of Replacement..................... Stationary sets ............................ Conclusion .............................. B. General Algebra Structures and homomorphisms Structures............................... Homomorphisms .......................... Categories .............................. Congruences and quotients .................... Trees and lattices Trees.................................. Lattices ................................ Ideals and filters........................... Prime ideals and ultrafilters .................... Atomic lattices and partition rank ................. Universal constructions Terms and term algebras ...................... Direct and reduced products.................... Directed limits and colimits .................... Equivalent diagrams......................... Links and dense functors...................... Accessible categories Filtered limits and inductive completions ............ Extensions of diagrams....................... Presentable objects ......................... Accessible categories ........................ vi Contents Topology Open and closed sets ........................ Continuous functions........................ Hausdorff spaces and compactness ................ te Product topology ........................ Dense sets and isolated points................... Spectra and Stoneduality ...................... Stone spaces and Cantor-Bendixson rank ............ Classical Algebra Groups ................................ Group actions ............................ Rings ................................. Modules................................ Fields ................................. Ordered fields ............................ C. First-Order Logicand its Extensions First-order logic Infinitary first-order logic ..................... Axiomatisations ........................... teories................................ Normal forms ............................ Translations ............................. Extensions of first-order logic ................... Elementary substructures and embeddings Homomorphisms and embeddings................ Elementary embeddings ...................... te teorem of Löwenheim and Skolem ............. vii Contents te Compactness teorem .................... Amalgamation ............................ Types and type spaces Types ................................. Type spaces.............................. Retracts................................ Local type spaces .......................... Stable theories ............................ Back-and-forth equivalence Partial isomorphisms........................ Hintikka formulae .......................... Ehrenfeucht-Fraïssé games..................... κ-complete back-and-forth systems................ te theorems of Hanf and Gaifman ................ General model theory Classifying logical systems..................... Hanf and Löwenheim numbers .................. te teorem of Lindström ..................... Projective classes........................... Interpolation ............................. Fixed-point logics.......................... D. Axiomatisation and Definability Quantifier elimination Preservation theorems ....................... Quantifier elimination ....................... Existentially closed structures ................... Abeliangroups............................ viii Contents Fields ................................. Products and varieties Ultraproducts ............................ te theoremofKeisler and Shelah ................ Reduced products and Horn formulae .............. Quasivarieties ............................ te teoremofFeferman and Vaught .............. O-minimal structures Ordered topological structures .................. O-minimalgroups and rings.................... Cell decompositions......................... E. Classical Model teory Saturation Homogeneous structures...................... Saturated structures......................... Projectively saturated structures.................. Pseudo-saturated structures .................... Definability and automorphisms Definability in projectively saturated models .......... Imaginary elements and canonical parameters ......... Galois bases ............................. Elimination of imaginaries..................... Weak elimination of imaginaries ................. ix Contents Prime models Isolated types............................. te Omitting Typesteorem ................... Prime and atomic models ..................... Constructible models........................ -categorical theories ℵ -categorical theories and automorphisms........... Bacℵ k-and-forth arguments in accessible categories....... Fraïssé limits............................. Zero-one laws ............................ Indiscernible sequences Ramsey teory ........................... Ramsey teory for trees ...................... Indiscernible sequences....................... te independence and strict order properties.......... Functors and embeddings Localfunctors ............................ Word constructions......................... Ehrenfeucht-Mostowski models.................. Abstract elementary classes Abstract elementary classes .................... Amalgamation and saturation ................... Limits of chains ........................... Categoricity and stability ...................... F. Independence and Forking x Contents Geometries Dependence relations........................ Matroids and geometries...................... Modular geometries......................... Strongly minimal sets........................ Vaughtian pairs and the teorem of Morley ........... Ranks and forking Morley rank and ∆-rank ...................... Independence relations....................... Preforking relations......................... Forking relations........................... Simple theories Dividing and forking ........................ Simple theories and the tree property ............... teories without the independence property Honest definitions.......................... Lascar invariant types........................ »i -Morley sequences ........................ Dp-rank ................................ teories without the array property te array property .......................... Forking anddividing ........................ te Independence teorem .................... G. Geometric Model teory xi Contents Stable theories Definable types............................ Forking in stable theories...................... Stationary types ........................... te multiplicity of a type ...................... Morley sequences in stable theories................ te stability spectrum ....................... Models of stable theories Isolation relations .......................... Constructions ............................ Prime models ............................ at»-constructible models...................... Strongly independent stratifications................ Representations ........................... Recommended Literature Symbol Index Index xii Part A. Set teory . Basic set theory . Sets and classes In mathematics there are basically two ways to define the objects under consideration. On the one hand, one can explicitly construct themfrom already known objects. For instance, the rational numbers and the real numbers are usually introduced in this way. On the other hand, one can take the axiomatic approach, that is, one compiles a list of desired prop- erties and one investigates any object meeting these requirements. Some well known examples are groups, fields, vector spaces, and topological spaces. Since set theory is meant as foundation of mathematics there are no more basic objects available in terms of which we could define sets. terefore, we will follow the axiomatic approach. We will present a list of six axioms and any object satisfying all of them will be called a model of set theory. Such a model consists of two parts: () a collection S of objects that we will call sets, and () some method which, given two sets a and b, tells us whether a is an element of b. We will not care what exactly the objects in S are or how this method looks like. For example, one could imagine a model of set theory
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