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3 Sets, Relations, and Functions

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.