New PDF release: Abstract Set Theory

By Thoralf Skolem

ISBN-10: 026800000X

ISBN-13: 9780268000004

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Everything must be conceived in relation to D as it is supposed to be by the axioms, and we must abandon the idea that the axioms shall yield an absolute notion of "set" as in Cantor's theory. That M is not ~ N means in the axiomatic theory that there is in D no set F of pairs (m,n), meM, neN, yielding a one-to-one correspondence between M and N. But that does not mean that we cannot find such a set at all. There might be such a set, but outside D. In this way there might be a one-to-one correspondence between the Zermelo number series consisting of the elements 0, {0}, {{0}}, ....

Theorem 43. Every inductive infinite set is Dedekind infinite. Proof. That the set u is_inductive infinite means that there exists a set x of subsets of u such that uex in spite of the circumstance that Oex and whenever yex & zeu, we have y U {z}ex. It is clear that there is no subset of u occurring as a greatest element of x. Now let us assume the principle of choice, that we have a function f of the subsets y of u such that always f(y)ey. Then we can define a g(y) for all yex thus: g(y) = f(u-y).

Of course M U N is cofinal with A. An arbitrary sequence S in M U N without last element is either such that from a certain point on all elements belong to M say, then the limit is in M; or there are always greater elements both in M and in N, and then there is a common limit in M and N. Theorem 21. If M and N are bands of A and A is as already indicated without last element, but not cofinal with a;, then M n N is a band of A. Proof. We assume that after a certain a0e M there are no common elements in M and N.

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Abstract Set Theory by Thoralf Skolem

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