Pairwise incompatible generic families

Abstract:

Let $M$ be a countable model of ${\mathbf {Z}}{{\mathbf {F}}^ - }$. There exists a family $\mathcal {F}$ of ${2^{{2^\omega }}}$ models of ${\mathbf {Z}}{{\mathbf {F}}^ - }$ each obtained from $M$ by adjoining an $M$-generic family of ${2^\omega }$ Cohen reals, such that no two distinct models in $\mathcal {F}$ have a common extension to a model of ${\mathbf {Z}}{{\mathbf {F}}^ - }$ with the same ordinals.