index of set theory

1 Basic Notions

1. set theory

2. set

3. subset

4. union

5. power set

6. generalized Cartesian product

7. transitive set

8. criterion for a set to be transitive

9. Cartesian product

10. proof of the associativity of the symmetric difference operator

11. proper subset

12. an example of mathematical induction

13. principle of finite induction

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

15. de Morgan’s laws

16. de Morgan’s laws for sets (proof)

2 Functions and Relations

1. antisymmetric

2. example of antisymmetric

3. argument

4. constant function

5. equivalence class

6. direct image

7. domain

8. fibre

9. fix (transformation action)

10. function

11. function graph

12. identity map

13. inclusion mapping

14. invariant

15. inverse image

16. irreflexive

17. left function notation

18. right function notation

19. level set

20. mapping

21. mapping of period $n$ is a bijection

22. operation

23. operations on relations

24. partial function

25. partial mapping

26. period of mapping

27. properties of a function

28. properties of functions

29. quasi-inverse of a function

30. range

31. reflexive relation

32. relation

33. restriction of a function

34. set difference

35. symmetric difference

36. symmetric relation

37. the inverse image commutes with set operations

38. transformation

39. transitive

40. transitive closure

41. transitive relation

42. choice function

43. one-to-one function from onto function

2.1 Order Relations

1. poset

2. maximal element

3. minimal element

4. visualizing maximal elements

5. cofinality

6. another definition of cofinality

7. chain

8. antichain

9. branch

10. tree (set theoretic)

11. example of tree (set theoretic)

12. proof that $\mathrm{\Omega }$ has the tree property

13. filtration

14. well ordered set

3 Cardinals and Ordinals

1. $\kappa$-complete

2. additively indecomposable

3. aleph numbers

4. algebraic numbers are countable

5. all algebraic numbers in a sequence

6. another proof of cardinality of the rationals

7. beth numbers

8. Cantor normal form

9. Cantor’s diagonal argument

10. Cantor’s theorem

11. cardinal arithmetic

12. cardinal exponentiation under GCH

13. cardinal number

14. cardinal successor

15. cardinality

16. cardinality of a countable union

17. cardinality of disjoint union of finite sets

18. cardinality of the continuum

19. cardinality of the rationals

20. classes of ordinals and enumerating functions

21. club

22. club filter

23. countable

24. countably infinite

25. finite

26. finite character

27. fixed points of normal functions

28. Fodor’s lemma

29. Hilbert’s hotel

30. if $A$ is infinite and $B$ is a finite subset of $A,$ then $A\setminus B$ is infinite

31. König’s theorem

32. limit cardinal

33. natural number

34. normal (ordinal) function

35. open and closed intervals have the same cardinality

36. ordinal arithmetic

37. ordinal number

38. pigeonhole principle

39. proof of pigeonhole principle

40. another proof of pigeonhole principle

41. proof of Cantor’s theorem

42. proof of fixed points of normal functions

43. proof of Fodor’s lemma

44. proof of the existence of transcendental numbers

45. proof of theorems in additively indecomposable

46. proof that countable unions are countable

47. proof that the rationals are countable

48. proof of Schroeder-Bernstein theorem

49. Schroeder-Bernstein theorem

50. stationary set

51. subsets of countable sets are countable

52. thin set

53. successor

54. successor cardinal

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

56. law of trichotomy

57. partitions less than cofinality

58. Aronszajn tree

59. example of Aronszajn tree

60. Suslin tree

61. Erdős-Rado theorem

62. uncountable owned by yark

63. uniqueness of cardinality

64. Veblen function

65. von Neumann integer

66. von Neumann ordinal

67. weakly compact cardinal

68. weakly compact cardinals and the tree property

69. inductive set

70. inaccessible cardinals

4 Axiomatic Formulation

1. axiom of choice

2. axiom of countable choice

3. axiom of determinacy

4. axiom of extensionality

5. axiom of infinity

6. axiom of pairing

7. axiom of power set

8. axiom of union

9. axiom schema of separation

10. continuum hypothesis

11. generalized continuum hypothesis

12. equivalence of Zorn’s lemma and the axiom of choice

13. Hausdorff’s maximum principle

14. Kuratowski’s lemma

15. maximality principle

16. permutation model

17. Tukey’s lemma

18. $𝒰$-small

19. proof of Tukey’s lemma

20. proof of Zermelo’s postulate

21. proof of Zermelo’s well-ordering theorem

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

23. relation as union of functions

24. Selector

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

26. well-ordering principle implies axiom of choice

27. Martin’s axiom

28. Martin’s axiom and the continuum hypothesis

29. Martin’s axiom is consistent

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

31. Zermelo’s postulate

32. Zermelo’s well-ordering theorem

33. Zorn’s lemma

34. example of universe

35. example of universe of finite sets

36. proof of properties of universe

37. Tarski’s axiom

38. universe

39. von Neumann-Bernays-Goedel set theory

40. chain condition

41. composition of forcing notions

42. composition preserves chain condition

43. equivalence of forcing notions

44. forcing

45. forcing relation

46. forcings are equivalent if one is dense in the other

47. FS iterated forcing preserves chain condition

48. iterated forcing

49. iterated forcing and composition

50. partial order with chain condition does not collapse cardinals

51. proof of partial order with chain condition does not collapse cardinals

52. proof that forcing notions are equivalent to their composition

53. Boolean valued model

54. complete partial orders do not add small subsets

55. proof of complete partial orders do not add small subsets

56. Levy collapse

57. $\mathrm{◇}$ is equivalent to $\mathrm{♣}$ and continuum hypothesis

58. proof of $\mathrm{◇}$ is equivalent to $\mathrm{♣}$ and continuum hypothesis

59. clubsuit

60. diamond

61. combinatorial principle

Title	index of set theory
Canonical name	IndexOfSetTheory
Date of creation	2013-03-22 16:40:32
Last modified on	2013-03-22 16:40:32
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	20
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 03E30