Axiom of Regularity

Set Theory · Axiom Academy

LESSON Axiom of Regularity (Foundation) Every non-empty set has a disjoint element Every non-empty set A contains at least one element x that is disjoint from A. This means x ∩ A = ∅. In other words: if A is non-empty, there exists some x ∈ A such that no element of x is also in A. This element x is called a "minimal" or "∈-minimal" element. One immediate consequence: No set can contain itself. If we had x ∈ x, then the set x would have no disjoint element, violating Regularity. The only element of x is x itself, and x ∩ x = x ≠ ∅ (since x ∈ x). This contradiction shows x ∈ x is impossible. 3. No Infinite Descending Chains Regularity also prevents infinite descending ∈-chains like ... ∈ x₃ ∈ x₂ ∈ x₁ ∈ x₀. If such a chain existed, the set x₀, x₁, x₂, ... would have no disjoint element (each xᵢ contains xᵢ₊₁ which is also in the set). This violates Regularity, so such chains cannot exist.

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