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# Completeness

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### Completeness

1. 1. COMPLETNESS CATEGORICITY COMPLETENESS SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO COMPLETENESS
2. 2. COMPLETNESS CATEGORICITY GÖDEL ERIK A. ANDREJKO COMPLETENESS
3. 3. COMPLETNESS CATEGORICITY COMPLETENESS THEOREM THEOREM (COMPLETENESS) |= =⇒ THEOREM (SOUNDNESS) =⇒ |= COROLLARY T is consistent if and only if T is satisﬁable. ERIK A. ANDREJKO COMPLETENESS
4. 4. COMPLETNESS CATEGORICITY ELEMENTARY SUBMODELS DEFINITION Let M ⊆ N. Then M is an elementary submodel of N if for all a ∈ M ¯ we have M |= ϕ(¯ ) ⇐⇒ N |= ϕ(¯ ) a a for all L -formulas ϕ. M N FACT Suppose that M ⊆ N. Let a ∈ M and suppose that ϕ(v ) is quantiﬁer ¯ ¯ free. Then M |= ϕ(¯ ) if and only if N |= ϕ(¯ ). a a ERIK A. ANDREJKO COMPLETENESS
5. 5. COMPLETNESS CATEGORICITY TARSKI VAUGHT TEST THEOREM (TARKSI VAUGHT TEST) Let M ⊆ N. Then M N if for all ϕ of the form ∃¯ ψ(¯ ) aa with ψ quantiﬁer free, M |= ϕ(¯ ) ⇐⇒ N |= ϕ(¯ ) a a ERIK A. ANDREJKO COMPLETENESS
6. 6. COMPLETNESS CATEGORICITY LOWENHEIM SKOLEM TARSKI THEOREM (DOWNWARD LOWENHEIM-SKOLEM-TARSKI) Let M be an L -structure, let κ be an inﬁnite cardinal with |L | ≤ κ ≤ |M|, 1 let A ⊆ M be any set with |A| = κ. 2 Then there exists a N ⊆ M with A ⊆ N, 1 |N| = κ, 2 N M. 3 ERIK A. ANDREJKO COMPLETENESS
7. 7. COMPLETNESS CATEGORICITY LOWENHEIM SKOLEM TARSKI THEOREM (UPWARD LOWENHEIM-SKOLEM-TARSKI) Let M be an L -structure and let κ ≥ |L | + |M| be a cardinal. Then there is an L -structure N with |N| = κ 1 M N. 2 ERIK A. ANDREJKO COMPLETENESS
8. 8. COMPLETNESS CATEGORICITY DEFINABILITY DEFINITION Let M be an L -structure. Let ϕ(¯ ) be a formula. Then a A = {x : M |= ϕ(¯ )} ¯ a is deﬁnable in M. ERIK A. ANDREJKO COMPLETENESS
9. 9. COMPLETNESS CATEGORICITY CATEGORICITY DEFINITION Let κ be an inﬁnite cardinal. Let Σ be an L -theory. Then Σ is κ-categorical if for every M, N with |M| = |N| = κ ∼ M |= Σ and N |= Σ =⇒ M = N ERIK A. ANDREJKO COMPLETENESS
10. 10. COMPLETNESS CATEGORICITY COMPLETE THEORIES THEOREM Let Σ be an L -theory. Suppose for some κ ≥ |L | for all M, N of size κ M≡N Then Σ is complete. COROLLARY Let Σ be κ-categorical for any κ ≥ |L |, then Σ is complete. ERIK A. ANDREJKO COMPLETENESS