Ineffability of \(\mathcal{P}_{\kappa} \lambda\) for \(\lambda\) with small cofinality
This paper is about ineffability, the Shelah property, and indescribability of \(\mathcal{P}_\kappa\lambda\). The author proves many interesting results about them. Let \(X\) be a subset of \(\mathcal{P}_\kappa\lambda\). \(X\) is said to be ineffable (respectively almost ineffable) if for every \(\langle a_x : x\in X\rangle\) with \(a_x\subseteq x\), there exists an \(A\subseteq\lambda\) such that \(\{x\in X : A\cap x=a_x\}\) is stationary (respectively unbounded). \(X\) has the Shelah property if for every \(\langle f_x : x\in X\rangle\) with \(f_x:x\rightarrow x\), there exists an \(f:\lambda\rightarrow\lambda\) such that for every \(y\in\mathcal{P}_\kappa\lambda\), \(\{x\in X : f\restriction y=f_x\restriction y\}\) is unbounded. Define \(\text{NIn}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ is not ineffable}\}\), \(\text{NAIn}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ is not almost ineffable}\}\), and \(\text{NSh}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ does not have the Shelah property}\}\). All of them form ideals over \(\mathcal{P}_\kappa\lambda\). \textit{D. M. Carr} [Z. Math. Logik Grundlagen Math. 31, 393--401 (1985; Zbl 0559.03033)] proved \(\text{NSh}_{\kappa \lambda}\subseteq\text{NIn}_{\kappa \lambda} \subseteq\text{NAIn}_{\kappa \lambda}\). Here are some of the results the author proves: (1) these three ideals are always strongly normal (\textit{D. M. Carr} [loc. cit.] proved this assuming \(\text{cf}(\lambda)\geq\kappa\), but it was open without the assumption); (2) if \(\mathcal{P}_\kappa\lambda\) has the Shelah property and \(\text{cf}(\lambda)<\kappa\), then \(\lambda^{{<}\kappa}=\lambda^+\); (3) if \(\lambda\) is strong limit with \(\text{cf}(\lambda)<\kappa\), then \(\text{NSh}_{\kappa \lambda}=\text{NIn}_{\kappa \lambda} =\text{NAIn}_{\kappa \lambda}\); (4) if \(\text{cf}(\lambda)\geq\kappa\), then, except the trivial case, \(\text{NSh}_{\kappa \lambda}\subsetneq\text{NIn}_{\kappa \lambda} \subsetneq\text{NAIn}_{\kappa \lambda}\); and (5) it is consistent relative to the existence of a Woodin cardinal that \(\text{NSh}_{\kappa \lambda}=\text{NIn}_{\kappa \lambda} =\text{NAIn}_{\kappa \lambda}\) are precipitous. He also proves that \(\Pi_1^1\)-indescribability of \(\mathcal{P}_\kappa\lambda\) is much stronger than ineffability when \(2^{\lambda}=\lambda^{{<}\kappa}\). This is in contrast to the result of \textit{Y.~Abe} [Arch. Math. Logic 37, No. 4, 261--272 (1998; Zbl 0907.03022)] that if \(\text{cf}(\lambda)\geq\kappa\), then \(\text{NSh}_{\kappa \lambda}=\Pi_{\kappa \lambda}\) where \(\Pi_{\kappa \lambda}\) denotes the set of all subsets of \(\mathcal{P}_{\kappa}\lambda\) that are not \(\Pi^1_1\)-indescribable.
scientific article; zbMATH DE number 5345244
scientific article