On shattering, splitting and reaping partitions. (English) Zbl 0896.03036

The author deals with cardinal invariants for the family of all partitions of \(\omega\). Especially he investigates dualizations of some well-known cardinal invariants for the power set of \(\omega\). So he considers the dual-shattering, dual-splitting and dual-reaping numbers. He shows that it is consistent with ZFC that the dual-shattering cardinal is greater than \(\omega_1\). This is done using dual-Mathias forcing.
Reviewer: M.Weese (Potsdam)


03E05 Other combinatorial set theory
03E35 Consistency and independence results
03C25 Model-theoretic forcing
05A18 Partitions of sets
Full Text: DOI arXiv


[1] Balcar, Fund. Math. 110 pp 11– (1980)
[2] Balcar, Topology and its Applications 41 pp 133– (1991)
[3] and , Set Theory: The structure of the real line. A.k. Peters, Wellesley 1995.
[4] Bell, Fund. Math. 114 pp 149– (1981)
[5] Carlson, Adv. in Math. 53 pp 265– (1984)
[6] , and , Dualizations of van Douwen diagram. Preprint.
[7] Multiple Forcing. Cambridge University Press, Cambridge 1987.
[8] Set Theory. Academic Press, London 1978.
[9] Set Theory, an Introduction to Independence Proofs. North-Holland Publ. Comp., Amsterdam 1983. · Zbl 0534.03026
[10] Matet, J. Symbolic Logic 51 pp 12– (1986)
[11] Piotrowski, Proc. Amer. Math. Soc. 101 pp 156– (1987)
[12] Plewnc, Fund. Math. 127 pp 127– (1986)
[13] Shelah, Contemporary Math. 31 pp 183– (1984)
[14] Partition numbers. Preprint.
[15] The integers and topology. In: Handbook of Set-theoretic Topology ( and , eds.), North-Holland Publ. Comp., Amsterdam 1990, pp. 111–167.
[16] Small uncountable cardinals and topology. In: Open Problems in Topology ( and , eds.) North-Holland Publ. Comp., Amsterdam 1990, pp. 195–218.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.