By A.A. Fraenkel, Y. Bar-Hillel
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Additional info for Foundations of Set Theory
However, all sets formed in this way contain just two members. To obtain sets of a more general character we have to look for another procedure. e. the construction of the union and the intersection of given sets, which are also the fundamental operations of Boolean algebra ( T , pp. 26, 109, 142). For our axiomatic purpose it will suffice to postulate the performability of one of these processes, and we choose union. According to our program, the sets whose union is to be formed shall not be taken arbitrarily but shall be members of a set the existence of which has already been ensured.
For every disjointed set t there exists the set whose members are just the sets 4) which contain a single member from each member of t. 1) Some authors use A, or 0. I n Theory the notation ‘0’(the same as for the number 0) is used. b’ (inner product), customary in France and Poland, is used. 3) One may carry out this proof without using Axiom 111. e. that for d E t , d # c, we have d , = c,( = S,). 4) These subsets of Ut are called ‘complexes’in T , p. 120. - Mostly the notion of Cartesian product is extended to the general case where the members of t need not be disjoint; then either a function-concept or ordered sets are utilized.
THEAXIOMOF CHOICE 1. Formulation of the Axiom. Its Introduction into Mathematics. After having obtained those subsets of a given set which are determined by a definite predicate we raise the question whether possibly other subsets, not obtained in this way, may be conceived and admitted; and if so, how far such subsets are necessary for developing set theory. The present section deals with an axiom which yields such subsets. Since t is disjointed all these single members are different. In a more elaborate form: any member y of t certainly does, or does not, satisfy, with respect to a given subset x of U t , the condition that the intersection x n y is a unit-set; the subset of those y E t which do satisfy this condition exists by Axiom V, and its being equal to t is the desired condition on x .