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Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted  aussonderung  but there are set theories ( e.g ., NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all (low) ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no set of  all  ordinals is obtained thereby. This “recurrence problem” is discussed.

Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF †