Philosophy Lexicon of Arguments

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Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle.
 
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Ba I 86
Axioms of the Set Theory/Halmos/Basieux: 1) extensionality axiom: two sets are only equal iff they have the same elements - 2) selection axiom: for every set A and every condition (or property) E(x) there is a set B, whose elements are exactly every x of A, for which E(x) applies - 3) pairing axiom: for every two sets there is always one set that contains those two as elements - 4) combination axiom: for every set system there is a set that contains all elements that belong to at least one set of the given system - 5) power set axiom: for every quantity there is a set system that contains all the subsets of the given set among its elements - 6) infinity axiom: there is a set that contains the empty set and with each of its elements also its successor - 7) choice axiom: the Cartesian product of a (non-empty) system of non-empty sets is non-empty - 8) replacement axiom: S(a,b) be a statement of the kind that for each element a of a set A the set {b I S (a,b)} can be formed. Then there is a function F with domain A such that F(a) = {b I S(a,b)} for every a in A -
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Ed. Martin Schulz, access date 2017-05-26