< κ. These properties assert that κ cannot be reached from below. The first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement. Just as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals. Zermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals. If κ is a strongly inaccessible cardinal, then transfinite induction proves |Vα| < κ for all α < κ: α = 0: |V0| = 0. For α+1: |Vα+1| = |Vα ∪ P(Vα)| ≤ |Vα| + 2 |Vα| = 2 |Vα| < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible. For limit α: |Vα| = |∪ξ < α Vξ| ≤ sup {|Vξ| : ξ < α} < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible. Proof of Theorem 1: A set X enters Vκ through P(Vα) for some α < κ, so X ⊆ Vα. Since |Vα| < κ, we obtain |X| < κ. Conversely: If a class X has |X| < κ, let β = sup {rank(x): x ∈ X}. Because κ is strongly inaccessible, |X| < κ and rank(x) < κ for all x ∈ X imply β = sup {rank(x): x ∈ X} < κ. Since rank(x) ≤ β for all x ∈ X, we have X ⊆ Vβ+1, so X ∈ Vβ+2 ⊆ Vκ. Therefore, X ∈ Vκ. Proof of Theorem 2: |Vκ| = |∪α < κ Vα| ≤ sup {|Vα| : α < κ}. Let β be this supremum. Since each ordinal in the supremum is less than κ, we have β ≤ κ. Assume β < κ. Then there is a cardinal λ such that β < λ < κ; for example, let λ = 2|β|. Since λ ⊆ Vλ and |Vλ| is in the supremum, we have λ ≤ |Vλ| ≤ β. This contradicts β < λ. Therefore, |Vκ| = β = κ. The sets and classes of Vκ satisfy all the axioms of NBG. Limitation of size doctrine The limitation of size doctrine is a heuristic principle that is used to justify axioms of set theory. It avoids the set theoretical paradoxes by restricting the full (contradictory) comprehension axiom schema: to instances "that do not give sets 'too much bigger' than the ones they use." If "bigger" means "bigger in