Axioms

• Is Consistency an Axiom of Mathematics?
  Consistency is usually a property of a formal mathematical theory, not a universal axiom; a theory can nevertheless include a formal consistency statement.
• 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.