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Frequently, logical formulas are interpreted in a domain whose objects are of
the datatype *set (Menge)*. The importance of this domain stems from its
universality: virtually all other types of objects that occur in mathematical
work (relations, functions, numbers, arrays, lists, trees, databases, ...)
can with the help of a few basic constructions be modelled as sets. The
properties of these objects are then determined entirely by the properties of
sets; the theory of sets thus provides the building material for most other
theories.

Intuitively, a set is a collection of elements. However, since sets shall
serve as a fundamental kind of objects, there is no point in asking what
(other object) a set *is* (if we could answer this question, we would
have a more fundamental kind of object). Sets are therefore not defined by
what they *are* but by what one *knows* about (respectively can
*do* with) them. In other words, the domain of sets is characterized by
various *axioms (Axiome)*, i.e., propositions that are stipulated
to be true.

A common axiomatization of *set theory (Mengenlehre)* is due to Zermelo
and Fraenkel; this form of set theory is called ZF set theory. We will not
list all ZF axioms but focus on their consequences for practical work.

Author: Wolfgang Schreiner

Last Modification: October 4, 1999