1

ByDr. Saqib Hussain

Introduction to Measure Theory

MTH 426

2

Ordinal Numbers

Lecture # 12

MTH 426

3

Previous Lecture’s Review

4

5

• Similar Sets

• Ordinal numbers

Lecture’s Outline

6

Theorem:

Proof:

7

Comparison of well ordered sets:

8

Theorem:

Proof:

9

10

11

Theorem:

Proof:

12

Theorem:

Proof:

13

Theorem:

Proof:

14

Theorem:

Proof:

15

16

17

Theorem:

Proof:

18

Ordinal Numbers:

Cardinal number of a well ordered set is called its ordinal number.

19

Inequalities in ordinal numbers:

20

Theorem:

Proof:

21

Theorem:

Proof:

22

23

24

Ordinal addition

Ordinal multiplication

25

Remark:

26

27

28

29

Remark:

Ordinal multiplication is non commutative

30

31

32

Remark:

Ordinal multiplication is associative

33

34

35

Choice Function:

36

Cartesian Product:

37

Axiom of choice:

Cartesian product of non empty family of non empty sets is non empty

OrThere exists a choice function for any non empty family of non empty sets.

38

Axiom of choice:

Cartesian product of non empty family of non empty sets is non empty

OrThere exists a choice function for any non empty family of non empty sets.

39

Zermelo’s Postulate:

40

Theorem:

Proof:

Show that axiom of choice is equivalent to Zermelo’s postulate.

41

42

References:

1. Set Theory and Related Topics by Seymour Lipschutz. 2. Elements of Set Theory by Herbert B. Enderton