The Indestructability of the Order of the Indescribable Cardinals

Abstract We prove the following consistency results about indescribable cardinals which answer a question of A. Kanamori and M. Magidor (cf. [3]). Theorem 1.1 (m⩾2, n⩾2). CON(ZFC + ∃κ, κ′ (κ is Πmn indescribable, κ′ is σmn indescribable, and κ πmn + GCH). Theorem 5.1 (ZFC). Assuming the existence of σmn indescribable cardinals for all m F : {(m,n): m ⩾ 2, n } 1} → {0,1} there is a poset P F ∈ L[ F ] such that GCH holds in (L[ F ])P F and ⊢;⊩ L| F P F σ m n m n if F (m, n)=0, σ m n >π m n if F (m, n)=1, Theorem 1.1 extends the work begun in [2], and its proof uses an iterated forcing construction together with master condition arguments. By combining these techniques with some observations about small forcing and indescribability, one obtains the Easton-style result 5.1.