## Ineffability of $$\mathcal{P}_{\kappa} \lambda$$ for $$\lambda$$ with small cofinality.(English)Zbl 1154.03034

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 $$\mathcal{P}_\kappa\lambda$$. 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 (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 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.

### MSC:

 3e+55 Large cardinals 300000 Other combinatorial set theory

### Citations:

Zbl 0559.03033; Zbl 0907.03022
Full Text:

### References:

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