Wikipedia defines the notion of a pure set as follows:
a hereditary set (or pure set) is a set whose elements are all hereditary sets.
Why does this definition make sense? It seems to be circular.
Also, wikipedia says:
The inductive definition of hereditary sets presupposes that set membership is well-founded (i.e., the axiom of regularity), otherwise the recurrence may not have a unique solution.
Why does the definition sometimes not have a unique solution? Is the problem the existence or the uniqueness? Can you give an example of a situation where the recursive definition from above does not have a unique solution in a setting where we don't assume regularity?
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$\begingroup$The axiom of regularity implies that there is no infinite descending sequence of sets. That is, the recursion in the definition has a finite limit and is thus well-defined.
Without that axiom, you are into the realm of non-well founded set theory where sets can be elements of themselves (which would form an infinite loop in the recursive definition of hereditary sets).
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