By Moshe Machover

During this creation to set concept and good judgment, the writer discusses first order good judgment, and offers a rigorous axiomatic presentation of Zermelo-Fraenkel set concept. He comprises many methodological comments and factors, and demonstrates how the fundamental techniques of arithmetic will be diminished to set thought. He explains ideas 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 advent to common sense and set concept.

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A2 , . , an} to its proper subset B- {a 1 } - contrary to the induction hypothesis. Thus B cannot be a proper subset of A. 5. Theorem For any natural numbers n and m: (i) if m ~ n then m ~ n; (ii) if m ::1= n then m ::1= n. (WARNING. ) § 4. Addition 43 PROOF (i) Assume m ~ rt. Take n distinct objects ah a 2, ... , On (which exist by Prob. 3). Since {a~o a2 , •• • , am} is clearly a subset of {uh a2, .. , on}, we have m ~ n by Thm. 3. (ii) Let m =I= n. Without loss of generality we may assume m < n .

10. Theorem The class of all objects (the universe of discourse) and the class of all sets are proper classes. PROOF We saw in § 2 that Russell's class, {x : xis a set such that x 1t x} cannot be a set. Since Russell's class is included in the class of all sets, the latter cannot be a set by AS. The same applies to the universe of discourse. • ~-- 1 §3. 11. Definition If A is any class, we put UA =dr {x: x e y for some ye A}. UA is called the union class of A. 12. Axiom of Union set (AU) If A is a set then so is UA.

9 (a)= a , whereas by Def. 10 (() , a) = (0, a). We proceed to define the notions of cartesum product and cm·tesian power. 12. Definition (i) For any classes A to A 2 , . . , A,, not necessarily distinct, their cartesian product [in this order] is the class A1 X A2 X •· • X An =de {(x~o x2, ... 'Xn): Xt EAt. X2 E A2, ... 'Xn E An}, that is, the class of all n-tuples whose i-th component belongs to A; for i = 1, 2 , . , n . (ii) The n -th cartesian power of a class A is the cartesian product of A with itself n times: An =de A X A X · ·· X A, 26 2.