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单词 IndexOfSetTheory
释义

index of set theory


1 Basic Notions

  1. 1.

    set theoryMathworldPlanetmath

  2. 2.

    set

  3. 3.

    subset

  4. 4.

    union

  5. 5.

    power setMathworldPlanetmath

  6. 6.

    generalized Cartesian product

  7. 7.

    transitive set

  8. 8.

    criterion for a set to be transitiveMathworldPlanetmathPlanetmath

  9. 9.

    Cartesian product

  10. 10.

    proof of the associativity of the symmetric difference operator

  11. 11.

    proper subsetMathworldPlanetmathPlanetmath

  12. 12.

    an example of mathematical induction

  13. 13.

    principle of finite induction

  14. 14.

    principle of finite induction proven from the well-ordering principle for natural numbers

  15. 15.

    de Morgan’s laws

  16. 16.

    de Morgan’s laws for sets (proof)

2 Functions and Relations

  1. 1.

    antisymmetric

  2. 2.

    example of antisymmetric

  3. 3.

    argument

  4. 4.

    constant function

  5. 5.

    equivalence classMathworldPlanetmathPlanetmath

  6. 6.

    direct imagePlanetmathPlanetmath

  7. 7.

    domain

  8. 8.

    fibre

  9. 9.

    fix (transformationPlanetmathPlanetmath action)

  10. 10.

    function

  11. 11.

    function graph

  12. 12.

    identity map

  13. 13.

    inclusion mapping

  14. 14.

    invariant

  15. 15.

    inverse imagePlanetmathPlanetmath

  16. 16.

    irreflexiveMathworldPlanetmath

  17. 17.

    left function notation

  18. 18.

    right function notation

  19. 19.

    level setPlanetmathPlanetmath

  20. 20.

    mapping

  21. 21.

    mapping of period n is a bijection

  22. 22.

    operationMathworldPlanetmath

  23. 23.

    operations on relations

  24. 24.

    partial functionMathworldPlanetmath

  25. 25.

    partial mapping

  26. 26.

    period of mapping

  27. 27.

    properties of a function

  28. 28.

    properties of functionsPlanetmathPlanetmath

  29. 29.

    quasi-inverse of a function

  30. 30.

    range

  31. 31.

    reflexive relation

  32. 32.

    relationMathworldPlanetmathPlanetmath

  33. 33.

    restrictionPlanetmathPlanetmathPlanetmathPlanetmath of a function

  34. 34.

    set differenceMathworldPlanetmath

  35. 35.

    symmetric differenceMathworldPlanetmath

  36. 36.

    symmetric relationMathworldPlanetmath

  37. 37.

    the inverse image commutes with set operations

  38. 38.

    transformation

  39. 39.

    transitive

  40. 40.

    transitive closureMathworldPlanetmathPlanetmath

  41. 41.

    transitive relation

  42. 42.

    choice function

  43. 43.

    one-to-one function from onto function

2.1 Order Relations

  1. 1.

    poset

  2. 2.

    maximal elementMathworldPlanetmath

  3. 3.

    minimal element

  4. 4.

    visualizing maximal elements

  5. 5.

    cofinality

  6. 6.

    another definition of cofinality

  7. 7.

    chain

  8. 8.

    antichainMathworldPlanetmath

  9. 9.

    branch

  10. 10.

    tree (set theoretic)

  11. 11.

    example of tree (set theoretic)

  12. 12.

    proof that Ω has the tree property

  13. 13.

    filtrationPlanetmathPlanetmath

  14. 14.

    well ordered set

3 Cardinals and Ordinals

  1. 1.

    κ-completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

  2. 2.

    additively indecomposable

  3. 3.

    aleph numbers

  4. 4.

    algebraic numbers are countable

  5. 5.

    all algebraic numbersMathworldPlanetmath in a sequenceMathworldPlanetmathPlanetmath

  6. 6.

    another proof of cardinality of the rationals

  7. 7.

    beth numbers

  8. 8.

    Cantor normal formMathworldPlanetmath

  9. 9.

    Cantor’s diagonal argument

  10. 10.

    Cantor’s theoremMathworldPlanetmath

  11. 11.

    cardinal arithmetic

  12. 12.

    cardinal exponentiation under GCH

  13. 13.

    cardinal numberMathworldPlanetmath

  14. 14.

    cardinal successor

  15. 15.

    cardinality

  16. 16.

    cardinality of a countableMathworldPlanetmath union

  17. 17.

    cardinality of disjoint union of finite sets

  18. 18.

    cardinality of the continuumMathworldPlanetmathPlanetmath

  19. 19.

    cardinality of the rationals

  20. 20.

    classes of ordinals and enumerating functions

  21. 21.

    club

  22. 22.

    club filter

  23. 23.

    countable

  24. 24.

    countably infiniteMathworldPlanetmath

  25. 25.

    finite

  26. 26.

    finite character

  27. 27.

    fixed points of normal functions

  28. 28.

    Fodor’s lemma

  29. 29.

    Hilbert’s hotel

  30. 30.

    if A is infiniteMathworldPlanetmath and B is a finite subset of A, then AB is infinite

  31. 31.

    König’s theorem

  32. 32.

    limit cardinal

  33. 33.

    natural numberMathworldPlanetmath

  34. 34.

    normal (ordinalMathworldPlanetmathPlanetmath) function

  35. 35.

    open and closed intervals have the same cardinality

  36. 36.

    ordinal arithmetic

  37. 37.

    ordinal number

  38. 38.

    pigeonhole principleMathworldPlanetmath

  39. 39.

    proof of pigeonhole principle

  40. 40.

    another proof of pigeonhole principle

  41. 41.

    proof of Cantor’s theorem

  42. 42.

    proof of fixed points of normal functions

  43. 43.

    proof of Fodor’s lemma

  44. 44.

    proof of the existence of transcendental numbersMathworldPlanetmath

  45. 45.

    proof of theorems in additively indecomposable

  46. 46.

    proof that countable unions are countable

  47. 47.

    proof that the rationals are countable

  48. 48.

    proof of Schroeder-Bernstein theorem

  49. 49.

    Schroeder-Bernstein theorem

  50. 50.

    stationary set

  51. 51.

    subsets of countable sets are countable

  52. 52.

    thin set

  53. 53.

    successorMathworldPlanetmathPlanetmathPlanetmath

  54. 54.

    successor cardinal

  55. 55.

    the Cartesian product of a finite number of countable sets is countable

  56. 56.

    law of trichotomy

  57. 57.

    partitions less than cofinality

  58. 58.

    Aronszajn tree

  59. 59.

    example of Aronszajn tree

  60. 60.

    Suslin tree

  61. 61.

    Erdős-Rado theorem

  62. 62.

    uncountable owned by yark

  63. 63.

    uniqueness of cardinality

  64. 64.

    Veblen function

  65. 65.

    von Neumann integer

  66. 66.

    von Neumann ordinal

  67. 67.

    weakly compact cardinal

  68. 68.

    weakly compact cardinals and the tree property

  69. 69.

    inductive setMathworldPlanetmath

  70. 70.

    inaccessible cardinalsMathworldPlanetmath

4 Axiomatic Formulation

  1. 1.

    axiom of choiceMathworldPlanetmath

  2. 2.

    axiom of countable choice

  3. 3.

    axiom of determinacy

  4. 4.

    axiom of extensionalityMathworldPlanetmath

  5. 5.

    axiom of infinityMathworldPlanetmath

  6. 6.

    axiom of pairing

  7. 7.

    axiom of power setMathworldPlanetmath

  8. 8.

    axiom of union

  9. 9.

    axiom schema of separation

  10. 10.

    continuum hypothesis

  11. 11.

    generalized continuum hypothesis

  12. 12.

    equivalence of Zorn’s lemma and the axiom of choice

  13. 13.

    HausdorffPlanetmathPlanetmath’s maximum principle

  14. 14.

    Kuratowski’s lemma

  15. 15.

    maximality principle

  16. 16.

    permutation model

  17. 17.

    Tukey’s lemma

  18. 18.

    𝒰-small

  19. 19.

    proof of Tukey’s lemma

  20. 20.

    proof of Zermelo’s postulateMathworldPlanetmath

  21. 21.

    proof of Zermelo’s well-ordering theorem

  22. 22.

    proof that a relation is union of functions if and only if AC

  23. 23.

    relation as union of functions

  24. 24.

    Selector

  25. 25.

    well-ordering principle for natural numbers proven from the principle of finite induction

  26. 26.

    well-ordering principle implies axiom of choice

  27. 27.

    Martin’s axiom

  28. 28.

    Martin’s axiom and the continuum hypothesis

  29. 29.

    Martin’s axiom is consistent

  30. 30.

    a shorter proof: Martin’s axiom and the continuum hypothesis

  31. 31.

    Zermelo’s postulate

  32. 32.

    Zermelo’s well-ordering theorem

  33. 33.

    Zorn’s lemma

  34. 34.

    example of universePlanetmathPlanetmath

  35. 35.

    example of universe of finite setsMathworldPlanetmath

  36. 36.

    proof of properties of universe

  37. 37.

    Tarski’s axiom

  38. 38.

    universe

  39. 39.

    von Neumann-Bernays-Goedel set theory

  40. 40.

    chain condition

  41. 41.

    composition of forcing notions

  42. 42.

    composition preserves chain condition

  43. 43.

    equivalence of forcing notions

  44. 44.

    forcingMathworldPlanetmath

  45. 45.

    forcing relation

  46. 46.

    forcings are equivalentMathworldPlanetmathPlanetmathPlanetmath if one is dense in the other

  47. 47.

    FS iterated forcing preserves chain condition

  48. 48.

    iterated forcing

  49. 49.

    iterated forcing and composition

  50. 50.

    partial order with chain condition does not collapse cardinals

  51. 51.

    proof of partial order with chain condition does not collapse cardinals

  52. 52.

    proof that forcing notions are equivalent to their composition

  53. 53.

    Boolean valued model

  54. 54.

    complete partial orders do not add small subsets

  55. 55.

    proof of complete partial orders do not add small subsets

  56. 56.

    Levy collapse

  57. 57.

    is equivalent to and continuum hypothesis

  58. 58.

    proof of is equivalent to and continuum hypothesis

  59. 59.

    clubsuit

  60. 60.

    diamondMathworldPlanetmath

  61. 61.

    combinatorial principle

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更新时间:2025/5/3 21:09:09