An Outline of Set Theory by James M. Henle

By James M. Henle

This booklet is designed to be used in a one semester problem-oriented direction in undergraduate set concept. the mix of point and structure is a little bit strange and merits a proof. in general, challenge classes are provided to graduate scholars or chosen undergraduates. i've got stumbled on, notwithstanding, that the event is both worthwhile to bland arithmetic majors. i exploit a contemporary amendment of R. L. Moore's well-known procedure constructed lately by means of D. W. Cohen [1]. in brief, during this new technique, initiatives are assigned to teams of scholars each one week. With the entire priceless the aid of the trainer, the teams whole their initiatives, conscientiously write a quick paper for his or her classmates, after which, within the unmarried weekly classification assembly, lecture on their effects. whereas the em­ phasis is at the scholar, the teacher is offered at each level to guarantee luck within the examine, to provide an explanation for and critique mathematical prose, and to teach the teams in transparent mathematical presentation. the subject material of set idea is particularly applicable to this form of path. for far of the booklet the items of research are ordinary and whereas the theorems are major and infrequently deep, it's the equipment and ideas which are most crucial. the need of rea­ soning approximately numbers and units forces scholars to come back to grips with the character of evidence, good judgment, and arithmetic. of their examine they event an identical dilemmas and uncertainties that confronted the pio­ neers.

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Logic and Set Theory 53 One consequence of this theorem is that Russell's paradox no longer haunts us. Actually, the paradox disappears even without using Regularity. Instead, we simply have a proof that R = {x Ix ¢ x} is not a set (because if it were, then we would have R E R iff R ¢ R -an impossibility). 5. Tricky again-apply Regularity to {A,B}. 7. Use Extension. PROJECT # 5. {a, b} fails, for example. If c =1= d, then we want

Theorem. 0 is an ordinal. to represent ordinals. 2. Theorem. N is an ordinal. 3. Theorem. If IX is an ordinal, then S(IX) is an ordinal. 4. Theorem. If IX is an ordinal and bE IX, then b is an ordinal. Definition. It is customary in set theory to write ()) for N. PROJECT # 25. 5. Theorem. 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.

X is pos iff -(x)z is neg. The product of two pos numbers is pos. 4. 14. 15. 6. 7. 61 The product of a pos and a neg is neg. The product of two neg numbers is pos. The sum of two positive numbers is positive. The sum of two negative numbers is negative. In proving some of these, you may have to go back (once again) to N and the definition of L is positive, then consider separately Case 1: Oz

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