| This thesis consists of two parts. In the fist part, considering the game theoretical aspect of generalized Cohen's forcing C(J) in dependence of ideal / on ω, Generalized a result of Sharp in Combinatorics on Ideals and Axiom A(The Journal of Symbolic Logic,1994), we argue that if J is a regular p* -ideal then C(J) do not satisfy Axiom A. We prove that the property satisfying Axiom A is not hereditary for dense subset. At the same time, it is pointed out that there is a gap in Collapsing of Cardinals in Generalized Cohen's Forcing(By M.Repicky, 1988).It is well known that partial order is closely relative to lattice. At a glance, the separativity of partial order and the distributiveness seems coherent.Do there on earth exist some interrelations well then? In the second part, we consider the intrinsic structure of the partition of ω, study the separativity of the orders ((ω)),≤) and discuss the distributivity of the lattices ((ω),≤). In the end, we give a more immediate and simple part proof of a_c ≤ a_s from Converse dual cardinals(By S.G.Zhang and J.brendle)... |
