The transcendence degree of the reals over certain set-theoretical subfields
Logic
2026-01-13 v2 Commutative Algebra
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 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 is also maximal. This answers a question of Kanovei and Schindler.
Cite
@article{arxiv.2412.00616,
title = {The transcendence degree of the reals over certain set-theoretical subfields},
author = {Azul Fatalini and Ralf Schindler},
journal= {arXiv preprint arXiv:2412.00616},
year = {2026}
}
Comments
We thank the reviewer that spotted a mistake in the argument of the proof Lemma 3.1. The proof has been replaced. Def 1.13 and Lemma 1.15 added, and other minor changes to improve the text. This is the last version, published in JSL