# index of set theory

## 1 Basic Notions

1. 1.
2. 2.

set

3. 3.

subset

4. 4.

union

5. 5.
6. 6.
7. 7.
8. 8.

criterion for a set to be transitive

9. 9.
10. 10.

proof of the associativity of the symmetric difference operator

11. 11.
12. 12.

an example of mathematical induction

13. 13.
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.
4. 4.
5. 5.
6. 6.
7. 7.

domain

8. 8.

fibre

9. 9.

fix (transformation action)

10. 10.
11. 11.
12. 12.
13. 13.
14. 14.
15. 15.
16. 16.
17. 17.
18. 18.
19. 19.
20. 20.
21. 21.

mapping of period $n$ is a bijection

22. 22.
23. 23.
24. 24.
25. 25.
26. 26.
27. 27.

properties of a function

28. 28.
29. 29.

quasi-inverse of a function

30. 30.

range

31. 31.
32. 32.
33. 33.

restriction of a function

34. 34.
35. 35.
36. 36.
37. 37.
38. 38.

transformation

39. 39.

transitive

40. 40.
41. 41.
42. 42.
43. 43.

### 2.1 Order Relations

1. 1.

poset

2. 2.
3. 3.

minimal element

4. 4.
5. 5.
6. 6.

another definition of cofinality

7. 7.

chain

8. 8.
9. 9.

branch

10. 10.

tree (set theoretic)

11. 11.

example of tree (set theoretic)

12. 12.

proof that $\Omega$ has the tree property

13. 13.
14. 14.

## 3 Cardinals and Ordinals

1. 1.

$\kappa$-complete

2. 2.
3. 3.
4. 4.
5. 5.
6. 6.

another proof of cardinality of the rationals

7. 7.
8. 8.

Cantor normal form

9. 9.

Cantor’s diagonal argument

10. 10.

Cantor’s theorem

11. 11.
12. 12.
13. 13.
14. 14.
15. 15.

cardinality

16. 16.

cardinality of a countable union

17. 17.
18. 18.

cardinality of the continuum

19. 19.

cardinality of the rationals

20. 20.
21. 21.

club

22. 22.
23. 23.

countable

24. 24.
25. 25.

finite

26. 26.
27. 27.
28. 28.

Fodor’s lemma

29. 29.

Hilbert’s hotel

30. 30.

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

31. 31.

König’s theorem

32. 32.
33. 33.
34. 34.

normal (ordinal) function

35. 35.

open and closed intervals have the same cardinality

36. 36.
37. 37.
38. 38.
39. 39.
40. 40.

another proof of pigeonhole principle

41. 41.

proof of Cantor’s theorem

42. 42.
43. 43.

proof of Fodor’s lemma

44. 44.
45. 45.
46. 46.
47. 47.

proof that the rationals are countable

48. 48.
49. 49.

Schroeder-Bernstein theorem

50. 50.
51. 51.
52. 52.

thin set

53. 53.
54. 54.
55. 55.

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

56. 56.
57. 57.
58. 58.

Aronszajn tree

59. 59.

example of Aronszajn tree

60. 60.
61. 61.

62. 62.

uncountable owned by yark

63. 63.
64. 64.
65. 65.
66. 66.
67. 67.
68. 68.

weakly compact cardinals and the tree property

69. 69.
70. 70.

## 4 Axiomatic Formulation

1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
10. 10.
11. 11.
12. 12.

equivalence of Zorn’s lemma and the axiom of choice

13. 13.
14. 14.

Kuratowski’s lemma

15. 15.
16. 16.
17. 17.

Tukey’s lemma

18. 18.

$\mathcal{U}$-small

19. 19.

proof of Tukey’s lemma

20. 20.

proof of Zermelo’s postulate

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.
24. 24.
25. 25.

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

26. 26.
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 universe

35. 35.

example of universe of finite sets

36. 36.
37. 37.

Tarski’s axiom

38. 38.

universe

39. 39.

von Neumann-Bernays-Goedel set theory

40. 40.
41. 41.
42. 42.
43. 43.
44. 44.
45. 45.
46. 46.

forcings are equivalent if one is dense in the other

47. 47.
48. 48.
49. 49.
50. 50.
51. 51.
52. 52.
53. 53.
54. 54.
55. 55.
56. 56.
57. 57.

$\Diamond$ is equivalent to $\clubsuit$ and continuum hypothesis

58. 58.

proof of $\Diamond$ is equivalent to $\clubsuit$ and continuum hypothesis

59. 59.
60. 60.
61. 61.
Title index of set theory IndexOfSetTheory 2013-03-22 16:40:32 2013-03-22 16:40:32 rspuzio (6075) rspuzio (6075) 20 rspuzio (6075) Definition msc 03E30