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

TL;DR

This paper extends known results about the transcendence degree of reals over ground models after Cohen forcing, showing it remains maximal even with multiple Cohen reals and subsets, answering a specific open question.

Contribution

It generalizes the maximal transcendence degree result from a single Cohen real to finitely many Cohen reals and their subsets, addressing an open problem.

Findings

Transcendence degree remains maximal after adding multiple Cohen reals.

The result applies to the extension over any proper subset of Cohen reals.

Answers a question posed by Kanovei and Schindler.

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.