标题： 实数在某些集合论子域上的超越次数 显示英文标题

标题： The transcendence degree of the reals over certain set-theoretical subfields

摘要： 这是一个众所周知的结果，即在添加一个 Cohen 实数后，实数在地面模型实数上的超越次数是连续统。 我们通过证明在力迫扩张中，实数相对于每个$X$的真子集给出的扩张中的实数的组合的超越次数也是最大的，从而将这一结果扩展到有限多个 Cohen 实数的集合$X$。 这回答了 Kanovei 和 Schindler 的一个问题。 显示英文摘要

摘要： 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.