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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.
Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

Author Concept Summary/Quotes Sources

Pierre Basieux on Set Theory - Dictionary of Arguments

Basieux I 86
Axioms of the Set Theory/Halmos(1)/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
>Selection axiom
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
>Power set.
6) infinity axiom: there is a set that contains the empty set and with each of its elements also its successor
>Infinity axiom
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.

1. Halmos, Paul (1974). Naive set theory, Santa Clara University.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
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