Ineffability of $¥mathscr{P}$_κλ for λ with small cofinality

Abstract

We study ineffability, the Shelah property, and indescribability of $¥mathscr{P}$_κλ when cf(λ) < κ. We prove that if λ is a strong limit cardinal with cf(λ) < κ then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over $¥mathscr{P}$_κλ are the same, and that it can be precipitous. Furthermore we show that Π¹₁-indescribability of $¥mathscr{P}$_κλ is much stronger than ineffability if 2^λ = λ^<κ.