theorem A.13 Completeness

open in the book · appendices/A-long-proofs.tex:1844 · p. 2797

Rests on

depends_on theorem A.12 Model existence ¶
- depends_on definition A.9 The term structure ¶
  - depends_on definition A.5 Maximal consistent ¶
    - depends_on definition 3.82 Consistency, completeness, soundness ¶
      - depends_on definition 3.80 Formal system ¶
      - depends_on definition 3.7 Negation ¶
  - depends_on equation A.112 eq:app-comp-Tstar ¶
- depends_on equation A.112 eq:app-comp-Tstar ¶ ↺
- depends_on lemma A.11 Truth lemma ¶
  - depends_on definition A.9 The term structure ¶ ↺
  - depends_on lemma A.7 Behaviour of a maximal consistent set ¶
    - depends_on definition A.5 Maximal consistent ¶ ↺
    - depends_on proposition 3.22 Algebra of propositions ¶
      - depends_on definition 3.17 Logical equivalence ¶
      - depends_on definition 3.6 Truth table ¶
      - proves proof ch:01-logic-sets@proof-4 ¶
    - proves proof app:A-long-proofs@proof-5 ¶
  - depends_on lemma A.10 lem:app-comp-welldefined ¶
    - depends_on definition A.9 The term structure ¶ ↺
    - depends_on definition 3.58 Equivalence relation ¶
    - depends_on lemma A.7 Behaviour of a maximal consistent set ¶ ↺
    - depends_on theorem 3.60 Equivalence classes partition the set ¶
      - depends_on definition 3.59 Equivalence class ¶
      - depends_on definition 3.58 Equivalence relation ¶ ↺
      - depends_on definition 3.36 Disjoint sets ¶
      - depends_on equation 3.47 eq:set-double-inclusion ¶
      - proves proof ch:01-logic-sets@proof-11 ¶
    - proves proof app:A-long-proofs@proof-7 ¶
  - depends_on lemma A.8 Adding witnesses preserves consistency ¶
    - depends_on definition 3.82 Consistency, completeness, soundness ¶ ↺
    - proves proof app:A-long-proofs@proof-6 ¶
  - proves proof app:A-long-proofs@proof-8 ¶
- proves proof app:A-long-proofs@proof-9 ¶
depends_on theorem A.4 Soundness ¶
- depends_on axiom 3.34 Principle of induction ¶
- depends_on definition 3.14 Tautology ¶
- depends_on equation 3.10 eq:log-conditional ¶
- proves proof app:A-long-proofs@proof-3 ¶
proves proof app:A-long-proofs@proof-10 ¶

Supports

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Neighborhood

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declared and complete
partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Model existence	declared	appendices/A-long-proofs.tex:1848
`depends_on`	→	Soundness	declared	appendices/A-long-proofs.tex:1848
`proves`	←	app:A-long-proofs@proof-10	declared	appendices/A-long-proofs.tex:1851