1. 1 By Dr. Saqib Hussain Introduction to Measure Theory MTH 426

2. 2 Ordinal Numbers Lecture # 12 MTH 426

3. 3 Previous Lecture’s Review

4. 4

5. 5 Similar Sets Ordinal numbers Lecture’s Outline

6. 6 Theorem: Proof:

7. 7 Comparison of well ordered sets:

8. 8 Theorem: Proof:

9. 9

10. 10

11. 11 Theorem: Proof:

12. 12 Theorem: Proof:

13. 13 Theorem: Proof:

14. 14 Theorem: Proof:

15. 15

16. 16

17. 17 Theorem: Proof:

18. 18 Ordinal Numbers: Cardinal number of a well ordered set is called its ordinal number.

19. 19 Inequalities in ordinal numbers:

20. 20 Theorem: Proof:

21. 21 Theorem: Proof:

22. 22

23. 23

24. 24 Ordinal addition Ordinal multiplication

25. 25 Remark:

26. 26

27. 27

28. 28

29. 29 Remark: Ordinal multiplication is non commutative

30. 30

31. 31

32. 32 Remark: Ordinal multiplication is associative

33. 33

34. 34

35. 35 Choice Function:

36. 36 Cartesian Product:

37. 37 Axiom of choice: Cartesian product of non empty family of non empty sets is non empty Or There exists a choice function for any non empty family of non empty sets.

38. 38 Axiom of choice: Cartesian product of non empty family of non empty sets is non empty Or There exists a choice function for any non empty family of non empty sets.

39. 39 Zermelo’s Postulate:

40. 40 Theorem: Proof: Show that axiom of choice is equivalent to Zermelo’s postulate.

41. 41

42. 42 References: 1. Set Theory and Related Topics by Seymour Lipschutz. 2. Elements of Set Theory by Herbert B. Enderton