The distributivity numbers of finite products of P(ω)/fin

Saharon Shelah; Otmar Spinas

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 1, page 81-93
  • ISSN: 0016-2736

Abstract

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Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o. ( P ( ω ) / f i n ) n , is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).

How to cite

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Shelah, Saharon, and Spinas, Otmar. "The distributivity numbers of finite products of P(ω)/fin." Fundamenta Mathematicae 158.1 (1998): 81-93. <http://eudml.org/doc/212304>.

@article{Shelah1998,
abstract = {Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.$(P(ω)/fin)^n$, is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).},
author = {Shelah, Saharon, Spinas, Otmar},
journal = {Fundamenta Mathematicae},
keywords = {forcing; Boolean algebras},
language = {eng},
number = {1},
pages = {81-93},
title = {The distributivity numbers of finite products of P(ω)/fin},
url = {http://eudml.org/doc/212304},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Shelah, Saharon
AU - Spinas, Otmar
TI - The distributivity numbers of finite products of P(ω)/fin
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 1
SP - 81
EP - 93
AB - Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.$(P(ω)/fin)^n$, is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).
LA - eng
KW - forcing; Boolean algebras
UR - http://eudml.org/doc/212304
ER -

References

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  1. [Ba] J. E. Baumgartner, Iterated forcing, in: Surveys in Set Theory, A. R. D. Mathias (ed.), London Math. Soc. Lecture Note Ser. 8, Cambridge Univ. Press, Cambridge, 1983, 1-59. 
  2. [BaPeSi] 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. [Go] M. Goldstern, Tools for your forcing construction, in: Israel Math. Conf. Proc. 6, H. Judah (ed.), Bar-Han Univ., Ramat Gan, 1993, 305-360. Zbl0834.03016
  4. [GoJoSp] M. Goldstern, M. Johnson and O. Spinas, Towers on trees, Proc. Amer. Math. Soc. 122 (1994), 557-564. Zbl0809.03035
  5. [GoReShSp] M. Goldstern, M. Repický, S. Shelah and O. Spinas, On tree ideals, ibid. 123 (1995), 1573-1581. Zbl0823.03027
  6. [JuSh] H. Judah and S. Shelah, Souslin forcing, J. Symbolic Logic 53 (1988), 1188-1207. 
  7. [Mt] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), 59-111. 
  8. [Shb] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, 1982. 
  9. [ShSp] S. Shelah and O. Spinas, The distributivity number of P(ω)/fin and its square, Trans. Amer. Math. Soc., to appear. 

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