Pairwise incompatible generic families

Abstract

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