Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{ +++} \leq \lambda$, $\kappa^{ <\kappa}=\kappa$ and $2^{ \kappa}= \kappa^+$, and $\eta$ is an ordinal with $\kappa^+\leq \eta <\kappa^{ ++}$ and $cf(\eta) = \kappa^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = \eta + 1$, - the cardinality of the $\alpha$th level of $B$ is $\kappa$ for every $\alpha <\eta$, - and the cardinality of the $\eta$th level of $B$ is $\lambda$ Especially, $\<{ \omega}\>_{ { \omega}_1}\concatenation \<{ \omega}_3\>$ and $\<{ \omega}_1\>_{ { \omega}_2}\concatenation \<{ \omega}_4\>$ can be cardinal sequences of superatomic Boolean algebras.
Categories: math.LO