basic constructions:
strong axioms
further
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An element of a set is a thing which “belongs to,” or “is an element of,” that set.
The circularity of this definition is unavoidable in foundational set theories in which “set” is an undefined term. In “definitional” set theories, where “set” is defined in terms of something else, elements are likewise defined in terms of the same “something else.”
If sets (or setoids) are regarded as the semantics of some type theory, then an element of a set is the interpretation of a term of some type.