  In this article we investigate the dual-shattering cardinal H, the
dual-splitting cardinal S and the dual-reaping cardinal R, which are
dualizations of the well-known cardinals h (the shattering cardinal, also known
as the distributivity number of P(omega) modulo finite, s (the splitting
number) and r (the reaping number). Using some properties of the ideal J of
nowhere dual-Ramsey sets, which is an ideal over the set of partitions of
omega, we show that add(J)=cov(J)=H. With this result we can show that H >
omega_1 is consistent with ZFC and as a corollary we get the relative
consistency of H > t, where t is the tower number. Concerning S we show that
cov(M) is less than or equal to S (where M is the ideal of the meager sets).
For the dual-reaping cardinal R we get p is less than or equal to R, which is
less than or equal to r (where p is the pseudo-intersection number) and for a
modified dual-reaping number R' we get that R' is less than or equal to d
(where d is the dominating number). As a consistency result we get R < cov(M).

Logic