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arXiv · 2512.02725

Transcendence degrees over mutually generic extensions

Jonathan Schilhan
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Abstract

Let $G_0$,..., $G_{n-1}$ be mutually generic over $V$, each $G_i$ adding at least one new real over $V$. We show that the transcendence degree of the reals of $V[G_0, \dots, G_{n-1}]$ is maximal (of size continuum) over the field generated by reals coming from models $V[ G_i : i \in a]$, for a proper subset $a$ of $n$. This answers a question of Fatalini and Schindler.

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Jonathan Schilhan. 2025-12-02. Transcendence degrees over mutually generic extensions. https://arxiv.org/abs/2512.02725

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