A *$\kappa$-tower* in $\mathbb{N}^\mathbb{N}$ is a sequence 
$\langle a\_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$
that is $\le^\*$-increasing with $\alpha$ 
and has no $\le^\*$-upper bound.


Piotr Szewczak and I need a reference for the consistency of the existence of a $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ for some (any) cardinal number $\kappa>\mathfrak{b}$.


It seems that this consistency (and much more) is established in
[Peter Dordal's paper on towers](http://www.sciencedirect.com/science/article/pii/0168007289900389), but we guess there is a more elementary and earlier reference for this specific result?


The motivation comes from selection principles (products of Menger spaces).



Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$
A theorem of Hechler says that, given any poset in which every countable subset has a strict upper bound, you can arrange for that poset to be cofinal in the $\leq^\*$ ordering of $\mathbb N^{\mathbb N}$. Apply that to the poset $\omega\_1\times\omega\_2$ (ordered componentwise). Then $\mathfrak b=\aleph\_1$, but $\mathbb N^{\mathbb N}$ also contains a tower of length $\omega\_2$.


