By James M. Henle

This ebook is designed to be used in a one semester problem-oriented direction in undergraduate set thought. the mix of point and structure is a bit strange and merits a proof. typically, challenge classes are provided to graduate scholars or chosen undergraduates. i've got came upon, even though, that the event is both important to bland arithmetic majors. i take advantage of a contemporary amendment of R. L. Moore's well-known approach constructed in recent times via D. W. Cohen [1]. in brief, during this new technique, initiatives are assigned to teams of scholars every one week. With the entire worthwhile the aid of the teacher, the teams whole their tasks, conscientiously 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 out there at each level to guarantee good fortune within the learn, to provide an explanation for and critique mathematical prose, and to teach the teams in transparent mathematical presentation. the subject material of set thought is in particular applicable to this sort of direction. for a lot of the e-book the gadgets of research are prevalent and whereas the theorems are major and infrequently deep, it's the equipment and ideas which are most vital. 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 identical dilemmas and uncertainties that confronted the pio­ neers.

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Since we can't subtract in N, we rewrite this as a +N d = b +N c. Now to expand Z to Q, we wish to acquire division. We can think of Q as ordered pairs (a, b), a, bE Z, which will represent division problems, alb. Of course b can't beO. Again, (a,b) and (c,d) should represent the same number iff alb = cld. Since we can't divide in Z, we have to rewrite this. PROJECT Definition. We define a relation (a,b) ~ (c,d) ~ on Z x (Z\{Oz}) by: iff a·z d = b· z c. Show this is an equivalence relation. It won't be easy!

Is the right cancellation law true? ) If +0 is not commutative, we also ask if the left cancellation law for +0 is true. Determine whether or not ·0 has the properties of commutativity or associativity. Does it have an identity? Do the cancellation laws hold? Is ·0 distributive over +o? Notation. It is standard practice to write our well-ordering on the ordinals. 0( < Pfor 0( E p, since E is 31 6. 13. 14. Theorem on Recursive Definitions. If f is any function, then there is afunction 9 such that for all ordinals a: g(a) = a iff a = f({g(P)IP < a}).

32 + 2 . 3 + 2 = 80 2 . 44 + 2 . 4 2 + 2 . 4 + 2 = 554 2· 44 + 2 . 4 2 + 2 . 5 2 + 2·5 + 1 = 6311 2· 55 + 2· 52 + 2· 5 = 6310 2 . 6 6 + 2 . 6 2 + 2 . 6 = 93396 2· 66 + 2 . 62 + 6 + 5 = 93395 (-1, rewrite) ('3' to '4's) ( -1, rewrite) The next few numbers are: 1647195, 33554571, 774841151, 20000000211, and 570623341475. How can we possibly get to O? Definition. For each n < Q) Q), 2::::;; n, we define a function Sn from as follows: Sn(O) = 0 Sn(k . (t) if k < n, Q) to 47 10. Goodstein's Theorem where ko, ...

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