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

Saharon Shelah; Otmar Spinas

ArticleOriginal scientific text

Title

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

Authors ¹, ²

Affiliations

Department of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem, Israel
Mathematik, ETH-Zentrum, 8092 Zürich, Switzerland

Abstract

Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.

{(P \frac{ω}{f} \in)}^{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).

[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.
[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.
[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.
[GoJoSp] M. Goldstern, M. Johnson and O. Spinas, Towers on trees, Proc. Amer. Math. Soc. 122 (1994), 557-564.
[GoReShSp] M. Goldstern, M. Repický, S. Shelah and O. Spinas, On tree ideals, ibid. 123 (1995), 1573-1581.
[JuSh] H. Judah and S. Shelah, Souslin forcing, J. Symbolic Logic 53 (1988), 1188-1207.
[Mt] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), 59-111.
[Shb] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, 1982.
[ShSp] S. Shelah and O. Spinas, The distributivity number of P(ω)/fin and its square, Trans. Amer. Math. Soc., to appear.

Title

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

Affiliations

Abstract

Bibliography