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Model Theory

Max Euwe. Model Theory. Jouko Väänänen. Signature. A signature is a set L of predicate symbols P,Q,R,... function symbols f,g,h,... constant symbols c,d,e,. Arity function:. Structure.

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Model Theory

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  1. Max Euwe Model Theory Jouko Väänänen Jouko Väänänen: Model theory

  2. Signature • A signature is a set L of • predicate symbols P,Q,R,... • function symbols f,g,h,... • constant symbols c,d,e,... Arity function: Jouko Väänänen: Model theory

  3. Structure • A structure (or model), for a signature L is a non-empty set M, called the domain or universe of M, and: • A subset PM of Mn for every predicate symbol P in L of arity n • A function fM of Mn into M for every function symbol f in L of arity n • An element cM of M for every constant symbol c in L. Jouko Väänänen: Model theory

  4. Unary structure M Vocabulary consists of one unary predicate symbol P0. Examples of structures P0 Jouko Väänänen: Model theory

  5. Unary structure with three predicates divides the domain into up to 8 parts. M P0 P1 P2 Jouko Väänänen: Model theory

  6. Graphs • A graph consists of vertices and edges between the vertices as in: • In this picture vertices are blue, edges are red. • Vertices connected by an edge are neighbors. Jouko Väänänen: Model theory

  7. More graphs Jouko Väänänen: Model theory

  8. A binary relation on a set M is any subsety R of the Cartesian product MxM. If (a,b) is in R, we write aRb, for simplicity Relational structure (M,R) MxM R b (a,b) a Jouko Väänänen: Model theory

  9. A binary relation is Symmetric if aRb implies bRa Reflexive if always aRa Transitive if aRb and bRc imply aRc Antisymmetric if aRb implies not bRa Antrireflexive if never aRa R R Properties of relational structures Jouko Väänänen: Model theory

  10. Equivalence relation Jouko Väänänen: Model theory

  11. Homomorphism Jouko Väänänen: Model theory

  12. Homomorphism defined Jouko Väänänen: Model theory

  13. Natural numbers with the successor function 0 1 2 … 0 1 Jouko Väänänen: Model theory

  14. Composition of homomorphisms Jouko Väänänen: Model theory

  15. Embedding h Jouko Väänänen: Model theory

  16. Isomorphism Jouko Väänänen: Model theory

  17. Automorphism Jouko Väänänen: Model theory

  18. Isomorphism defined Jouko Väänänen: Model theory

  19. Properties • Composition of isomorphisms is an isomorphism • Inverse of an isomorphism is an isomorphism • Identity is an automorphism • Automorphisms form a group Jouko Väänänen: Model theory

  20. Monadic (unary) structure  P0 P0  Jouko Väänänen: Model theory

  21. Equivalence relation Isomorphism Jouko Väänänen: Model theory

  22. Equivalence relation Automorphism Jouko Väänänen: Model theory

  23. Total (linear)order is isomorphic to ((0,1),<) ((0,1),<) 0 Jouko Väänänen: Model theory

  24. Well-order … … … The isomorphism is unique! Jouko Väänänen: Model theory

  25. Dense total order Jouko Väänänen: Model theory

  26. Substructure Jouko Väänänen: Model theory

  27. Equivalent • M is a substructure of M’ • Inclusion idM:MM’ is an embedding Jouko Väänänen: Model theory

  28. Natural numbers with the successor function 0 1 2 … 0 1 2 1 2 … 0 2 … Jouko Väänänen: Model theory

  29. Integers with the successor function -2 -1 0 1 2 … … 0 1 2 -2 0 1 -2 -4 0 2 4 Jouko Väänänen: Model theory

  30. Generated substructure <X>M M Subset Generated substructure Jouko Väänänen: Model theory

  31. Generated substructure The idea of the proof Jouko Väänänen: Model theory

  32. The successor function again 0 1 2 … 0 1 2 What do these generate? What do these generate? What do these generate? 1 2 … 0 2 … Jouko Väänänen: Model theory

  33. Preliminary observation • Equivalent: • X = dom(A) for some substructure A of B • X contains cB for all cL and is closed under fB for each f  L • There is at most such A. • Proof: (a)(b): Follows from the definition of substructure. • (b)(a): Define A in the obvious way. The domain of A is X. cA=cB, fA=fB|Xn, RA=RB|Xn Jouko Väänänen: Model theory

  34. Proof of the Lemma • There is a unique smallest substructure A of B, such that Ydom(A). • It is denoted <Y>B • |<Y>B||Y|+|L| • <Y>B is countable, if Y and L are • Proof: Let Y0=Y{cB : cL}, Yn+1=Yn {fB(a) : fL, a1,…,anYn}, and Y*= nYn. By the previous lemma, Y* is the domain of a substructure A of B. By construction it is the smallest such. Jouko Väänänen: Model theory

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