By James M. Henle

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Additional resources for An Outline of Set Theory

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For IX, 13, ordinals, IX ~ 13 ~ IX E 13. 6. Theorem. E is a linear ordering on the ordinals. 7. Theorem. E is a well-ordering on the ordinals. 8. Theorem. If A is a set of ordinals, then UA is an ordinal and is the least upper bound of A. 9. Theorem. There is no largest ordinal. 10. Theorem. The collection of all ordinals is not a set. Definition. If an ordinal IX = S(f3) for some 13, then IX is a successor ordinal. If IX =f:. 0 is not a successor ordinal, it is a limit ordinal. PROJECT # 26. 11.

Ilwll. 3. Theorem. II w I ~ II Q II ~ II w II· 34 Part One PROJECT # 29. 4. Theorem. (The Shroeder-Bernstein Theorem). 5. Theorem. I w I = I £: I = ~ IIBII ~ I Q II· PROJECT # 30. 6. Theorem (Georg Cantor, 1874). I A I < I P(A) I for all sets A. 7. Theorem (Georg Cantor, 1874). I w I < IIIR II. Definition. An ordinal a is called a cardinal if 11[311 < I all for all [3 < a. It is customary to use the Hebrew letter ~ to represent cardinals. S. Theorem. If A is a set of cardinals, then UA is a cardinal.

As before, 7L and 0 are completely different, yet 7L can be embedded in O. Can the reader find a copy of 7L in O? PROJECT # 19. This one is difficult, primarily because there are many facts about