On absolutely divergent series

Sakaé Fuchino; Heike Mildenberger; Saharon Shelah; Peter Vojtáš

Fundamenta Mathematicae (1999)

  • Volume: 160, Issue: 3, page 255-268
  • ISSN: 0016-2736

Abstract

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We show that in the 2 -stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under c f ( ) = the two algebras are isomorphic [15].

How to cite

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Fuchino, Sakaé, et al. "On absolutely divergent series." Fundamenta Mathematicae 160.3 (1999): 255-268. <http://eudml.org/doc/212392>.

@article{Fuchino1999,
abstract = {We show that in the $ℵ_2$-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under $cf() = $ the two algebras are isomorphic [15].},
author = {Fuchino, Sakaé, Mildenberger, Heike, Shelah, Saharon, Vojtáš, Peter},
journal = {Fundamenta Mathematicae},
keywords = {-distributive Boolean algebra; Mathias forcing; absolutely divergent series; factor algebra of the power set algebra},
language = {eng},
number = {3},
pages = {255-268},
title = {On absolutely divergent series},
url = {http://eudml.org/doc/212392},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Fuchino, Sakaé
AU - Mildenberger, Heike
AU - Shelah, Saharon
AU - Vojtáš, Peter
TI - On absolutely divergent series
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 3
SP - 255
EP - 268
AB - We show that in the $ℵ_2$-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under $cf() = $ the two algebras are isomorphic [15].
LA - eng
KW - -distributive Boolean algebra; Mathias forcing; absolutely divergent series; factor algebra of the power set algebra
UR - http://eudml.org/doc/212392
ER -

References

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  1. [1] N. Alon, J. Spencer and P. Erdős, The Probabilistic Method, Wiley, 1992. 
  2. [2] B. Balcar, J. Pelant and P. Simon, The space of ultrafilters on N covered by nowhere dense sets, Fund. Math. 110 (1980), 11-24. Zbl0568.54004
  3. [3] T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), 209-213. Zbl0538.03042
  4. [4] T. Bartoszyński and M. Scheepers, Remarks on small sets related to trigonometric series, Topology Appl. 64 (1995), 133-140. Zbl0828.42004
  5. [5] J. Baumgartner, Iterated forcing, in: A. Mathias (ed.), Surveys in Set Theory, London Math. Soc. Lecture Note Ser. 8, Cambridge Univ. Press, 1983, 1-59. Zbl0524.03040
  6. [6] E. van Douwen, The integers and topology, in: K. Kunen and J. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland, 1984, 111-167. 
  7. [7] G. M. Fikhtengolz, Course of Differential and Integral Calculus, Nauka, Moscow, 1969 (in Russian). 
  8. [8] G. H. Hardy, Orders of Infinity. The 'Infinitärcalcül' of Paul du Bois Reymond, Cambridge Univ. Press, 1910. Zbl41.0303.01
  9. [9] F. Hausdorff, Summen von 1 Mengen, Fund. Math. 26 (1936), 241-255. 
  10. [10] T. Jech, Set Theory, Addison-Wesley, 1978. 
  11. [11] T. Jech, Distributive laws, in: D. Monk (ed.), Handbook of Boolean Algebras, North-Holland, 1989, 317-332. 
  12. [12] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, 1980. 
  13. [13] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, 1982. 
  14. [14] S. Shelah and O. Spinas, The distributivity numbers of P(ω)/fin and its square, Trans. Amer. Math. Soc., to appear. Zbl0943.03036
  15. [15] P. Vojtáš, Boolean isomorphism between partial orderings of convergent and divergent series and infinite subsets of ℕ, Proc. Amer. Math. Soc. 117 (1993), 235-242. Zbl0765.03023
  16. [16] P. Vojtáš, On ω* and absolutely divergent series, Topology Proc. 19 (1994), 335-348. Zbl0840.03033
  17. </REFERENCES> 

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