The indescribability 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[ges]2, n[ges]2). CON(ZFC + [there exists][kappa], [kappa]' ([kappa] is [Pi]mn indescribable, [kappa]' is [sigma]mn indescribable, and [kappa] CON(ZFC + [sigma] mn> [pi]mn + GCH).Theorem 5.1 (ZFC). Assuming the existence of [sigma]mn indescribable cardinals for all m {0,1} there is a poset P [set membership, variant] L[] such that GCH holds in (L[])P and 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.