Ineffability of P κ λ for λ with small cofinality

Abstract

We study ineffability, the Shelah property, and indescribability of $P_{\kappa }\lambda$ when $\text{cf}(\lambda )<\kappa$. We prove that if $\lambda$ is a strong limit cardinal with $\text{cf}(\lambda )<\kappa$ then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over $P_{\kappa }\lambda$ are the same, and that it can be precipitous. Furthermore we show that ${\Pi }_1^1$-indescribability of $P_{\kappa }\lambda$ is much stronger than ineffability if $2^{\lambda }={\lambda }^{<\kappa }$.