Using Koszmider's strongly unbounded functions, we show the following
consistency result:
Suppose that κ,λ are infinite cardinals such that κ+++≤λ, κ<κ=κ and 2κ=κ+, and η
is an ordinal with κ+≤η<κ++ and cf(η)=κ+.
Then, in some cardinal-preserving generic extension there is a superatomic
Boolean algebra B such that - ht(B)=η+1, - the cardinality of the
αth level of B is κ for every α<η, - and the
cardinality of the ηth level of B is λ Especially,
\_{{\omega}_1}\concatenation \$ and
\_{{\omega}_2}\concatenation \$ can be cardinal
sequences of superatomic Boolean algebras.Comment: 13 page