Axiom of Foundation

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The Axiom of Foundation is the most recent of the axioms of Zermelo-Fraenkel set theory to be added to the list. It states that for every set A the sequence:

  • A
  • A1 an element of A
  • A2 an element of A1
  • A3 an element of A2
  • etc.

must eventually stop. In other words, every set A has a bottom, or foundation. It is therefore impossible in Zermelo-Fraenkel set theory to define a set as A={A}. In other words, A cannot be defined to be an element of itself.

This axiom eliminates circular definitions from the logic of set theory and also quashes several paradoxes, a great achievement for logic.