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

top
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 c f ( ) = the two algebras are isomorphic [15].

How to cite

top

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

top
  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 ℵ 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> 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.