Elementary set theory

• Can a Set Be an Element of Itself?
  In ZF and ZFC, the Axiom of Foundation rules out x ∈ x. See the short proof, why the axiom matters, and how non-well-founded theories differ.
• What Does It Mean for a Set Not to Contain the Empty Set?
  A set that does not contain the empty set satisfies ∅ ∉ A; this differs from being nonempty and from the fact that ∅ is a subset of every set.