Transcendence degrees over mutually generic extensions

TL;DR

This paper proves that the transcendence degree of reals in a union of mutually generic extensions is maximal over any proper subset of those extensions, answering a question in set theory.

Contribution

It establishes the maximality of the transcendence degree over sub-collections of mutually generic extensions, providing a definitive answer to a previously open question.

Findings

Transcendence degree is maximal over sub-collections of generic extensions.

The result applies to models with added reals via mutually generic extensions.

Answers a specific open question in set theory about generic extensions.

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.