By Moshe Machover
ISBN-10: 0521474930
ISBN-13: 9780521474931
ISBN-10: 0521479983
ISBN-13: 9780521479981
During this advent to set idea and common sense, the writer discusses first order good judgment, and provides a rigorous axiomatic presentation of Zermelo-Fraenkel set idea. He comprises many methodological feedback and factors, and demonstrates how the fundamental suggestions of arithmetic might be diminished to set concept. He explains techniques and result of recursion concept in intuitive phrases, and reaches the limitative result of Skolem, Tarski, Church and G?del (the celebrated incompleteness theorems). for college students of arithmetic and philosophy, this ebook offers an exceptional creation to common sense and set thought.
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Extra info for Set Theory, Logic and Their Limitations
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2) it is easy to see that A' n B' = 0. Also, a bijection f from A ' to A is obtained by putting f {a, x) = x for every x e A; so IAI = IA'I. Similarly, IBI = IB'I. 3. Lemma Let A, B, A', B' be sets such that An B =A' n B' and IBI = IB'I. Then lA U Bl = lA' U B'l. = 0, IAI = lA'l PROOF Let f and g be bijections from A to A' and from B to B' respectively. Then it is clear that f U g is a bijection from A U B to A' u B'. 4. l. 5. Remarks (i) Def. 3. (ii) In the proof of Thm. 3 . We had there A= G U (A- G) and B = f(G] U (A- G), where the unions in both cases are between disjoint sets.
9, says that X {Bx IX E X} = C X D. So in this case we have IX{Bx lx E X}l = IC X Dl, which is what Def. 1 says xA. should be. § 6. 1. Definition Let A and B be any sets. Then map(A , B) = dt {f: f is a map from A to B}. 2. Rei1Ulrks (i) If f is any member of map(A, B) then f ~A x B, hence f is a member of P(A x B). Thus map(A, B) C P(A x B), and map(A, B) is a set. 10, as follows. Consider the indexed family § 6. Exponentiation; Cantor's Theorem 51 { D0 I a eA} such that Da = B for every a e A.
Rei1Ulrks (i) If f is any member of map(A, B) then f ~A x B, hence f is a member of P(A x B). Thus map(A, B) C P(A x B), and map(A, B) is a set. 10, as follows. Consider the indexed family § 6. Exponentiation; Cantor's Theorem 51 { D0 I a eA} such that Da = B for every a e A. 10- is , by Def. 9 equal to {f : f is a function such that do m f = A and fa e B for all a e A }. By Def. 1 this is exactly map(A, B) . 3. Definition For any cardinals ).. and JJ, we define JJ to the [po wer of} k where A ,I = lmap(A, B)l, and Bare sets such tha t lA I=)..
Set Theory, Logic and Their Limitations by Moshe Machover
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