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

Usuba, Toshimichi

J. Math. Soc. Japan 60, No. 3, 935-954 (2008).

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.

Reviewer: Tetsuya Ishiu (Oxford, Ohio)

MSC:

03E55	Large cardinals
03E05	Other combinatorial set theory

Keywords:

\(\mathcal{P}_\kappa \lambda\); ineffable; almost ineffable; Shelah property; indescribable; ideals

Citations:

Zbl 0559.03033; Zbl 0907.03022

References:

[1]	Y. Abe, Saturation of fundamental ideals on \({P}_ \kappa\lambda\), J. Math. Soc. Japan, 48 (1996), 511-524. · Zbl 0865.03038
[2]	Y. Abe, Combinatorial characterization of \(\Pi^ 1_ 1\)-indescribability in \(P_ \kappa\lambda\), Arch. Math. Logic, 37 (1998), 261-272. · Zbl 0907.03022
[3]	Y. Abe, Notes on subtlety and ineffability in \(P_ \kappa\lambda\), Arch. Math. Logic, 44 (2005), 619-631. · Zbl 1073.03024
[4]	J. E. Baumgartner, Ineffability properties of cardinals I, Infinite and finite sets Vol. I, Colloq. Math. Soc. Janos Bolyai, 10 , North-Holland, Amsterdam, 1975, pp.,109-130. · Zbl 0313.04002
[5]	J. P. Burgess, Forcing, Handbook of mathematical logic, (ed. J. Barwise), Studies in Logic and the Foundations of Mathematics, 90 , North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[6]	M. R. Burke, M. Magidor, Shelah’s \({\mathrm pcf}\) theory and its applications, Ann. Pure Appl. Logic, 50 (1990), 207-254. · Zbl 0713.03024
[7]	D. M. Carr, The minimal normal filter on \(P_{\kappa}\lambda\), Proc. Amer. Math. Soc., 86 (1982), 316-320. · Zbl 0497.03037
[8]	D. M. Carr, \(P_ \kappa\lambda\)-generalizations of weak compactness, Z. Math. Logik Grundlag. Math., 31 (1985), 393-401. · Zbl 0559.03033
[9]	D. M. Carr, The structure of ineffability properties of \(P_ \kappa\lambda\), Acta Math. Hungar., 47 (1986), 325-332. · Zbl 0612.03023
[10]	D. M. Carr, J.-P. Levinski, D. H. Pelletier, On the existence of strongly normal ideals over \(P_ \kappa\lambda\), Arch. Math. Logic, 30 (1990), 59-72. · Zbl 0711.03019
[11]	P. Erdös, A. Hajnal, On a problem of B. Jonsson, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 14 (1966), 19-23. · Zbl 0171.26501
[12]	N. Goldring, The entire NS ideal on \(P _ \gamma \mu\) can be precipitous, J. Symbolic Logic, 62 (1997), 1161-1172. · Zbl 0941.03051
[13]	T. J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic, 5 (1973), 165-198. · Zbl 0262.02062
[14]	T. J. Jech, Set theory, The third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[15]	C. A. Johnson, On ideals and stationary reflection, J. Symbolic Logic, 54 (1989), 568-575. · Zbl 0703.03030
[16]	C. A. Johnson, Some partition relations for ideals on \(P_ \kappa\lambda\), Acta Math. Hungar., 56 (1990), 269-282. · Zbl 0733.03039
[17]	S. Kamo, Remarks on \(P_ \kappa \lambda\)-combinatorics, Fund. Math., 145 (1994), 141-151. · Zbl 0807.03029
[18]	A. Kanamori, The higher infinite, Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. · Zbl 0813.03034
[19]	S. Shelah, Cardinal arithmetic, Oxford Logic Guides, 29 , Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
[20]	R. Solovay, Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., XXV , Univ. California, Berkeley, Calif., 1971), pp.,365-372. · Zbl 0317.02083
[21]	J. Tryba, Rowbottom-type properties and a cardinal arithmetic, Proc. Amer. Math. Soc., 96 (1986), 661-667. · Zbl 0615.03038