The Indestructability of the Order of the Indescribable Cardinals

Hauser, K., The indescribability of the order of the indescribable cardinals, Annals of Pure and Applied Logic 57 (1992) 45-91. 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 22, n 22). CON(ZFC+ 3~, K’ (K is Ii: indescribable, K’ is XF indescribable, and K < K’)) j CON(ZFC + fl> n: + GCH). ‘Theorem 5.1 (ZFC). Assuming the existence of ZT indescribable cardinals for all m < o and n < w and given a function 9: {(m, n): m 5 2, n 2 l} + {0, l}, there is a poset P* E L[.Fj such that GCH holds in (L[%j)pF and ,tLIsI ( c < jc? if Wn, n) = 0, Q fl>Ic,” if s(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.