## Complete quotient Boolean algebras

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- by Akihiro Kanamori and Saharon Shelah PDF
- Trans. Amer. Math. Soc.
**347**(1995), 1963-1979 Request permission

## Abstract:

For $I$ a proper, countably complete ideal on the power set $\mathcal {P}(x)$ for some set $X$, can the quotient Boolean algebra $\mathcal {P}(X)/I$ be complete? We first show that, if the cardinality of $X$ is at least ${\omega _3}$, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal $I$ is a (nontrivial) ideal over a cardinal $\kappa$ which is ${\kappa ^ + }$-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over $\kappa = {\omega _1}$ relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: ${2^{{\omega _1}}} = {\omega _3}$ and there is an ideal ideal $I$ over ${\omega _1}$ such that $\mathcal {P}({\omega _1})/I$ is complete. (The cardinality assertion implies that there is no ideal over ${\omega _1}$ which is ${\omega _2}$-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)## References

- James E. Baumgartner,
*Iterated forcing*, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR**823775**, DOI 10.1017/CBO9780511758867.002 - James E. Baumgartner and Alan D. Taylor,
*Saturation properties of ideals in generic extensions. I*, Trans. Amer. Math. Soc.**270**(1982), no. 2, 557–574. MR**645330**, DOI 10.1090/S0002-9947-1982-0645330-7 - M. Foreman, M. Magidor, and S. Shelah,
*Martin’s maximum, saturated ideals, and nonregular ultrafilters. I*, Ann. of Math. (2)**127**(1988), no. 1, 1–47. MR**924672**, DOI 10.2307/1971415 - Moti Gitik and Saharon Shelah,
*Cardinal preserving ideals*, J. Symbolic Logic**64**(1999), no. 4, 1527–1551. MR**1780068**, DOI 10.2307/2586794 - Thomas Jech,
*Set theory*, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**506523** - Jussi Ketonen,
*Some combinatorial principles*, Trans. Amer. Math. Soc.**188**(1974), 387–394. MR**332481**, DOI 10.1090/S0002-9947-1974-0332481-5 - Kenneth Kunen,
*Some applications of iterated ultrapowers in set theory*, Ann. Math. Logic**1**(1970), 179–227. MR**277346**, DOI 10.1016/0003-4843(70)90013-6 - R. S. Pierce,
*Distributivity in Boolean algebras*, Pacific J. Math.**7**(1957), 983–992. MR**89180**, DOI 10.2140/pjm.1957.7.983 - S. Shelah,
*A weak generalization of MA to higher cardinals*, Israel J. Math.**30**(1978), no. 4, 297–306. MR**505492**, DOI 10.1007/BF02761994
—, - Saharon Shelah,
*Iterated forcing and normal ideals on $\omega _1$*, Israel J. Math.**60**(1987), no. 3, 345–380. MR**937796**, DOI 10.1007/BF02780398 - Saharon Shelah,
*Proper and improper forcing*, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR**1623206**, DOI 10.1007/978-3-662-12831-2 - R. Sikorski,
*On an unsolved problem from the theory of Boolean algebras*, Colloq. Math.**2**(1949), 27–29. MR**40273**, DOI 10.4064/cm-2-1-27-29 - E. C. Smith Jr. and Alfred Tarski,
*Higher degrees of distributivity and completeness in Boolean algebras*, Trans. Amer. Math. Soc.**84**(1957), 230–257. MR**84466**, DOI 10.1090/S0002-9947-1957-0084466-4 - Robert M. Solovay,
*Real-valued measurable cardinals*, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR**0290961**

*Proper forcing*, Lecture Notes in Math., vol. 940, Springer-Verlag, Berlin, 1986.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**347**(1995), 1963-1979 - MSC: Primary 03E35; Secondary 03E40, 03E55, 06E05
- DOI: https://doi.org/10.1090/S0002-9947-1995-1282888-0
- MathSciNet review: 1282888