definition 3.7 Negation

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Supports

• depends_on definition 3.82 Consistency, completeness, soundness ¶
  • depends_on definition A.5 Maximal consistent ¶
    • depends_on definition A.9 The term structure ¶
      • depends_on lemma A.11 Truth lemma ¶
        • depends_on theorem A.12 Model existence ¶
      • depends_on lemma A.10 lem:app-comp-welldefined ¶
        • depends_on lemma A.11 Truth lemma ¶ ↺
      • depends_on theorem A.12 Model existence ¶ ↺
    • depends_on lemma A.6 Lindenbaum ¶
    • depends_on lemma A.7 Behaviour of a maximal consistent set ¶
      • depends_on lemma A.11 Truth lemma ¶ ↺
      • depends_on lemma A.10 lem:app-comp-welldefined ¶ ↺
  • depends_on lemma A.8 Adding witnesses preserves consistency ¶
    • depends_on lemma A.11 Truth lemma ¶ ↺
  • depends_on theorem 3.84 Gödel's completeness theorem, 1930 ¶
    • depends_on theorem A.30 Church, Turing ¶
  • depends_on theorem 3.87 Gödel, 1931 ¶
    • depends_on theorem A.29 Second incompleteness theorem ¶
    • depends_on theorem 3.88 Gödel's second incompleteness theorem ¶
  • depends_on theorem 3.91 Tarski's undefinability of truth ¶
• depends_on definition 3.35 Union, intersection, difference ¶
  • depends_on definition 6.3 Closed set ¶
    • depends_on definition 12.70 Graph; closed and closable operators ¶
      • depends_on proposition 12.73 The adjoint is always closed ¶
        • depends_on definition 12.78 Essential self-adjointness ¶
        • depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
    • depends_on definition 10.79 Parabolic boundary ¶
      • depends_on theorem 10.80 Parabolic maximum principle ¶
        • depends_on corollary 10.81 Uniqueness for the heat equation ¶
        • depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
    • depends_on theorem 6.12 Heine–Borel in $\R^{N}$ ¶
  • depends_on definition 6.1 Topological space ¶
    • depends_on definition A.529 Hausdorff; second countable; locally compact ¶
      • depends_on lemma A.530 Two elementary facts about compactness ¶
        • depends_on proposition A.532 Separation and countability of the quotient ¶
    • depends_on definition A.298 Half-space; smoothness on it ¶
      • depends_on definition A.299 Manifold with boundary ¶
        • depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
        • depends_on lemma A.307 Partition of unity on a compact manifold ¶
    • depends_on definition 11.33 Sigma-algebra ¶
      • depends_on definition 11.34 Measure; measure space ¶
        • depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
        • depends_on definition 11.35 Null set; almost everywhere ¶
        • depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
    • depends_on definition 6.6 Continuous map ¶
      • depends_on definition A.361 Covering map ¶
        • depends_on lemma A.366 Unique path lifting ¶
        • depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
        • depends_on theorem A.364 Covering homomorphisms ¶
        • depends_on theorem A.363 Monodromy ¶
      • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
        • depends_on definition A.580 Weyl operator ¶
        • depends_on definition 12.109 Weyl system ¶
        • depends_on lemma A.255 Smoothed vectors lie in the domain ¶
        • depends_on lemma A.581 Composition law ¶
        • depends_on lemma A.582 Joint strong continuity ¶
        • depends_on proposition A.256 Density ¶
        • depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
        • depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
        • depends_on theorem A.253 Stone ¶
        • depends_on theorem 12.66 Stone ¶
        • depends_on theorem 25.34 Stone–von Neumann ¶
      • depends_on definition 6.7 Homeomorphism ¶
        • depends_on definition 13.50 Diffeomorphism ¶
        • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on lemma A.536 A slice is a chart domain downstairs ¶
        • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
        • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      • depends_on definition 6.15 Path-connected space ¶
        • depends_on definition A.362 Path homotopy and the fundamental group ¶
        • depends_on definition 6.17 Simply connected space ¶
        • depends_on proposition 6.16 prop:top-path-implies-connected ¶
      • depends_on definition 6.17 Simply connected space ¶ ↺
      • depends_on lemma A.505 A continuous partition of unity ¶
        • depends_on lemma A.506 Locality ¶
      • depends_on lemma A.531 The projection is open ¶
        • depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
        • depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
      • depends_on proposition 6.10 Continuous images of compact sets ¶
        • depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
      • depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
        • depends_on proposition 12.36 Boundedness is continuity ¶
    • depends_on definition 6.26 Open ball; metric topology ¶
      • depends_on definition A.76 Star-shaped set ¶
        • depends_on lemma A.77 Poincaré lemma, converse form ¶
      • depends_on lemma 6.30 Lebesgue number ¶
        • depends_on theorem 6.31 Compactness and sequential compactness ¶
      • depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶ ↺
    • depends_on definition 6.2 Open set ¶
      • depends_on definition A.361 Covering map ¶ ↺
      • depends_on definition A.141 Mollification ¶
        • depends_on proposition A.143 Reproduction identity ¶
      • depends_on definition A.134 Mean-value property ¶
        • depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
        • depends_on proposition A.143 Reproduction identity ¶ ↺
        • depends_on theorem A.135 Converse of the mean-value property ¶
      • depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
      • depends_on definition 13.108 Star-shaped domain ¶
        • depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
      • depends_on definition 10.1 Partial differential equation; order ¶
        • depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
        • depends_on definition 10.2 Linear, semilinear, quasilinear ¶
        • depends_on definition 10.24 Well-posed problem ¶
      • depends_on definition 6.3 Closed set ¶ ↺
      • depends_on definition 6.13 Connected space ¶
        • depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
        • depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
        • depends_on theorem 6.14 Intervals are connected ¶
      • depends_on definition 6.6 Continuous map ¶ ↺
      • depends_on definition 6.4 Neighbourhood ¶
      • depends_on definition 6.26 Open ball; metric topology ¶ ↺
      • depends_on definition 6.5 Open cover ¶
        • depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
        • depends_on definition 6.9 Compact set ¶
        • depends_on lemma 6.30 Lebesgue number ¶ ↺
      • … 2 more
  • depends_on proposition 3.54 Preimages respect the set operations ¶
  • depends_on proposition 3.37 De Morgan's laws for sets ¶
• depends_on proposition 3.19 Conditional as disjunction ¶
  • depends_on proposition 3.21 Contraposition ¶
• depends_on proposition 3.20 De Morgan's laws ¶
• depends_on theorem 3.75 Russell, 1901 ¶