Simplicity TheoryModel theory, a major branch of mathematical logic, plays a key role connecting logic and other areas of mathematics such as algebra, geometry, analysis, and combinatorics. Simplicity theory, a subject of model theory, studies a class of mathematical structures, called simple. The class includes all stable structures (vector spaces, modules, algebraically closed fields, differentially closed fields, and so on), and also important unstable structures such as the random graph, smoothly approximated structures, pseudo-finite fields, ACFA and more. Simplicity theory supplies the uniform model theoretic points of views to such structures in addition to their own mathematical analyses. This book starts with an introduction to the fundamental notions of dividing and forking, and covers up to the hyperdefinable group configuration theorem for simple theories. It collects up-to-date knowledge on simplicity theory and it will be useful to logicians, mathematicians and graduate students working on model theory. |
Contents
1 Introduction | 1 |
2 Dividing Forking and Simplicity | 12 |
3 Lascar Strong Types and Type Amalgamation | 45 |
4 Hyperimaginaries and Canonical Bases | 55 |
5 Elimination of Hyperimaginaries | 81 |
6 Constructing Simple Structures | 108 |
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1-based 3-independent A-indiscernible sequence a)-categorical acl(A algebraic amalgamation base arity assume Aut(M automorphism bdd(A canonical base Cb(p Claim clearly commensurate compactness complete type contradiction Corollary coset countable definable equivalence relation Definition divide division ring dom(D equivalence relation finite field following are equivalent fork formula geometric stability theory geometry group configuration Hence holds hyperdefinable subgroup hyperimaginaries imaginary tuple implies indiscernible infinite interbounded interdefinable isomorphism k-divides k-inconsistent Lascar left-generic Lemma Moreover Morley sequence non-forking extension non-trivial normal subgroup Note notion partial type pregeometry Proof Proposition quantifier elimination random graph realizing Remark resp satisfies Similarly simple theories slº solution set Stab(p stable theories strong types strongly minimal structures SU-rank SU(G SU(p subgroup of G subset suffices to show supersimple theories Suppose tp(a tree property tuple type amalgamation type p(x type-definable equivalence relation vector space