An uncountable regular strong-limit cardinal (κ>ℵ₀, cf(κ)=κ, ∀λ<κ: 2^λ<κ). V_κ ⊨ ZFC, so existence is not provable in ZFC (2nd incompleteness).
An uncountable regular strong-limit cardinal (κ>ℵ₀, cf(κ)=κ, ∀λ<κ: 2^λ<κ). V_κ ⊨ ZFC, so existence is not provable in ZFC (2nd incompleteness).