An Outline of Set Theory by James M. Henle

By James M. Henle

This ebook is designed to be used in a one semester problem-oriented direction in undergraduate set conception. the combo of point and layout is a bit of strange and merits a proof. mostly, challenge classes are provided to graduate scholars or chosen undergraduates. i've got chanced on, notwithstanding, that the event is both useful to boring arithmetic majors. i take advantage of a contemporary amendment of R. L. Moore's recognized strategy constructed in recent times via D. W. Cohen [1]. in brief, during this new process, initiatives are assigned to teams of scholars every one week. With the entire priceless the aid of the teacher, the teams whole their tasks, rigorously write a quick paper for his or her classmates, after which, within the unmarried weekly type assembly, lecture on their effects. whereas the em­ phasis is at the scholar, the teacher is on the market at each level to guarantee luck within the study, to provide an explanation for and critique mathematical prose, and to educate the teams in transparent mathematical presentation. the subject material of set conception is mainly acceptable to this form of path. for a lot of the publication the gadgets of research are generic and whereas the theorems are major and infrequently deep, it's the equipment and concepts which are most crucial. the need of rea­ soning approximately numbers and units forces scholars to come back to grips with the character of facts, common sense, and arithmetic. of their study they adventure an analogous dilemmas and uncertainties that confronted the pio­ neers.

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To show that {{a}, {a, b}} succeeds, you must show that whenever {{a}, {a, b}} = {{c}, {c, d} }, then a = c and b = d. 8. Show that A x B is a definable subset of P(P(A U B)). 9. Show that the range of f is a definable subset of U(U f). 10. Show that f PROJECT f D is a definable subset of f. # 6. 12. Use Comprehension. 13. Suppose that [ah n [bh =1= 0. Show first that aRb. Remember that to prove two sets x and y equal, you should show: (i) that every member of x is also a member of y, and (ii) that every member of y is also a member of x.

N. 10 which offers three equivalent forms of the Axiom of Choice. We then use AC to construct a system of numbers called the Hyperreal numbers (1J-lR). This system extends IR as IR extended (Jl and (Jl extended 71. IJ-lR contains both infinite numbers and infinitesimals. PROJECT # 39. 1. Theorem. The Axiom of Choice implies Zermelo's Theorem. 2. Theorem. Zermelo's Theorem implies Zorn's Lemma. PROJECT # 40. 3. Theorem. Zorn's Lemma implies the Axiom of Choice. 10. Definition. 0/1 £; P(ro) is a (non-principal) ultrafilter on ro iff (a) if A, BEo/1 then A n BEo/1, (b) if IIAII < ro then ro\AEo/1, and (c) for all A £; ro, either A or ro\A is in 0/1.

8. By induction on n: let cp(n) be the statement: 0 +1'\1 n = n. PROJECT #9. 9. Commutativity turns out to be much trickier than associativity. Try this: first prove a + 1'\1 1 = 1 + 1'\1 a for all a E N (by induction), then prove a +1'\1 b = b +1'\1 a from this (also by induction). 12. Use induction on p. 13. You don't need induction. 14. 11. 56 Part Two PROJECT # 11. 15. Once again, the commutative law is the most difficult. 1'\1 a = 0 for all a EN. 1'\1 a using induction on a. 1'\1 S(a) for all bEN.

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