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# Compactness

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### Compactness

1. 1. LANGUAGES MODELS COMPACTNESS THEOREM COMPACTNESS SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO COMPACTNESS
2. 2. LANGUAGES MODELS COMPACTNESS THEOREM LANGUAGES DEFINITION The logical symbols of propositional logic are: ∧, ∨, =⇒ , ¬ The objects of the logic are propositional letters: P1 , P2 , · · · e.g. ((P1 ∨ P2 ) =⇒ P17 ) =⇒ ¬(P2 ∧ P13 ) ERIK A. ANDREJKO COMPACTNESS
3. 3. LANGUAGES MODELS COMPACTNESS THEOREM LANGUAGES DEFINITION The logical symbols of ﬁrst order logic are: ∧, ∨, =⇒ , ¬, =, ∀, ∃ and variables (bound or unbound) x, y , z, · · · and additional symbols (constant, relational, functional) c1 , c2 , c3 , · · · R1 , R2 , R3 , · · · f , g, h, · · · e.g. ∀x∃y(f (x)R1 g(y )). ERIK A. ANDREJKO COMPACTNESS
4. 4. LANGUAGES MODELS COMPACTNESS THEOREM SPECIAL SYMBOLS DEFINITION 1 ∀∞ means for all but ﬁnitely many. ∃∞ means there exists inﬁnitely many. 2 ∃!x means there exists a unique x. 3 DEFINITION 1 Σ : ∃xϕ(x), ϕ(x) is q.f. 1 Π1 : ∀xϕ(x), ϕ(x) is q.f. 2 Σn+1 : ∃xϕ(x), ϕ(x) is Πn . 3 Πn+1 : ∀xϕ(x), ϕ(x) is Σn . 4 e.g. ∀x∃y[x > y ]. ERIK A. ANDREJKO COMPACTNESS
5. 5. LANGUAGES MODELS COMPACTNESS THEOREM FIRST ORDER LANGUAGE DEFINITION A ﬁrst order language L is a triple F, R, C F is a set of function symbols, 1 R is a set of relation symbols, 2 C is a set of constant symbols. 3 ERIK A. ANDREJKO COMPACTNESS
6. 6. LANGUAGES MODELS COMPACTNESS THEOREM STRUCTURES DEFINITION An L -structure M is a quadruple M = M, {f M }f ∈F , {R M }R∈R , {c M }c∈C M is a non empty set called the universe of M, 1 {f M }f ∈F is a set of functions, 2 {R M }R∈R is a set of relations, 3 {c M }c∈C ⊆ M. 4 ERIK A. ANDREJKO COMPACTNESS
7. 7. LANGUAGES MODELS COMPACTNESS THEOREM SENTENCES ∃v∀y[xRv ∨ f (y) = v ] DEFINITION ∀y or ∃v BOUND FREE otherwise DEFINITION An L -formula ϕ is called a sentence if all variables in ϕ are bound. ERIK A. ANDREJKO COMPACTNESS
8. 8. LANGUAGES MODELS COMPACTNESS THEOREM EMBEDDINGS DEFINITION σ : M → N is an L-embedding if σ is one-to-one and σ (f M (¯ )) = f N (σ (¯ )) for all functions f , a a 1 M ⇐⇒ σ (¯ ) ∈ R N for all relations R, a∈R ¯ a 2 σ (c M ) = c N for all constants c. 3 DEFINITION If σ is a bijective L -embedding, then σ is called an isomorphism. If ∼ there is a isomorphism between M and N then M = N. DEFINITION If there exists an L -embedding from M into N then M is called a substructure of N and N is called an extension of M. ERIK A. ANDREJKO COMPACTNESS
9. 9. LANGUAGES MODELS COMPACTNESS THEOREM |= AND DEFINITION First Order Propositional ∃ proof ∃ proof |= ∃ model ∃ valuation and |=. The completeness theorem relates ERIK A. ANDREJKO COMPACTNESS
10. 10. LANGUAGES MODELS COMPACTNESS THEOREM CONSISTENT DEFINITION A set of L -sentences is called a theory. DEFINITION An L -theory T is called consistent if there does not exists an L -sentence ϕ such that ϕ ∧ ¬ϕ T A consistent theory T is denoted Con(T). ERIK A. ANDREJKO COMPACTNESS
11. 11. LANGUAGES MODELS COMPACTNESS THEOREM ELEMENTARY EQUIVALENCE DEFINITION M and N are called elementarily equivalent, denoted M ≡ N, if M |= ϕ if and only if N |= ϕ for all L -sentences ϕ. FACT ∼ Suppose that M = N. Then M ≡ N. ERIK A. ANDREJKO COMPACTNESS
12. 12. LANGUAGES MODELS COMPACTNESS THEOREM THEORIES DEFINITION An L -theory T is any set of L -sentences. DEFINITION A theory is called satisﬁable if there is some model M |= T . DEFINITION Given any L -structure M Th(M) = {ϕ : M |= ϕ} ERIK A. ANDREJKO COMPACTNESS
13. 13. LANGUAGES MODELS COMPACTNESS THEOREM COMPACTNESS THEOREM THEOREM (Compactness Theorem) T is satisﬁable if and only if every ﬁnite subset of T is satisﬁable. ERIK A. ANDREJKO COMPACTNESS