The distributivity numbers of $\mathcal {P}(\omega )$/fin and its square
Authors:
Saharon Shelah and Otmar Spinas
Journal:
Trans. Amer. Math. Soc. 352 (2000), 2023-2047
MSC (1991):
Primary 03E05, 06E05
DOI:
https://doi.org/10.1090/S0002-9947-99-02270-9
Published electronically:
April 13, 1999
MathSciNet review:
1751223
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length $\omega _{2}$, the distributivity number of ${\mathcal {P}}(\omega )$/fin is $\omega _{2}$, whereas the distributivity number of r.o.$({\mathcal {P}}(\omega )$/fin)$^{2}$ is $\omega _{1}$. This answers a problem of Balcar, Pelant and Simon, and others.
- James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1β59. MR 823775, DOI https://doi.org/10.1017/CBO9780511758867.002
- Bohuslav Balcar, Jan Pelant, and Petr Simon, The space of ultrafilters on ${\bf N}$ covered by nowhere dense sets, Fund. Math. 110 (1980), no. 1, 11β24. MR 600576, DOI https://doi.org/10.4064/fm-110-1-11-24
- M. Goldstern and S. Shelah, Ramsey ultrafilters and the reaping numberβ${\rm Con}({\mathfrak r}<{\mathfrak u})$, Ann. Pure Appl. Logic 49 (1990), no. 2, 121β142. MR 1077075, DOI https://doi.org/10.1016/0168-0072%2890%2990063-8
- Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
- Jaime I. Ihoda and Saharon Shelah, $\Delta ^1_2$-sets of reals, Ann. Pure Appl. Logic 42 (1989), no. 3, 207β223. MR 998607, DOI https://doi.org/10.1016/0168-0072%2889%2990016-X
- Jaime I. Ihoda and Saharon Shelah, Souslin forcing, J. Symbolic Logic 53 (1988), no. 4, 1188β1207. MR 973109, DOI https://doi.org/10.2307/2274613
- Haim Judah and Otmar Spinas, Large cardinals and projective sets, Arch. Math. Logic 36 (1997), no. 2, 137β155. MR 1462204, DOI https://doi.org/10.1007/s001530050059
- A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59β111. MR 491197, DOI https://doi.org/10.1016/0003-4843%2877%2990006-7
- Mary Ellen Rudin, Partial orders on the types in $\beta N$, Trans. Amer. Math. Soc. 155 (1971), 353β362. MR 273581, DOI https://doi.org/10.1090/S0002-9947-1971-0273581-5
- Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
- S. Shelah, Paper 630, to appear.
- S. Shelah, O. Spinas, The distributivity numbers of finite products of ${\mathcal P}(\omega )$/fin, Fund. Math. 158 (1998), 81β93. .
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Additional Information
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem, Israel
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
Email:
shelah@math.huji.ac.il
Otmar Spinas
Affiliation:
Mathematik, ETH-Zentrum, 8092 ZΓΌrich, Switzerland
Email:
spinas@math.ethz.ch
Received by editor(s):
February 12, 1997
Received by editor(s) in revised form:
November 5, 1997
Published electronically:
April 13, 1999
Additional Notes:
The first author is supported by the Basic Research Foundation of the Israel Academy of Sciences; publication 494. The second author is supported by the Swiss National Science Foundation.
Article copyright:
© Copyright 2000
American Mathematical Society
