THE TRANSCENDENCE DEGREE OF THE REALS OVER CERTAIN SET-THEORETICAL SUBFIELDS

Abstract It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set X of finitely many Cohen reals, by showing that, in the forcing extension, the transcendence degree of the reals over a combination of the reals in the extension given by each proper subset of X is also maximal. This answers a question of Kanovei and Schindler [2].