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Transcendence degrees over mutually generic extensions

Jonathan Schilhan

TL;DR

Addresses the question of Fatalini and Schindler about the algebraic independence of reals in mutually generic forcing extensions. Proves that the transcendence degree of the reals in V[G_0,...,G_{n-1}] over the field generated by the reals from any proper subcollection is maximal, i.e., continuum. The proof combines De Bondt's really-closure trick (via a countable family of C^1 closures) with a novel non-similarity argument on perfect rectangles to prevent representation of new reals from intermediate models, first in the n=2 case and then in general n. The results extend known Cohen-forcing cases and contribute to the broader understanding of the algebraic structure of forcing extensions, with potential further explicit witness constructions.

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.

Transcendence degrees over mutually generic extensions

TL;DR

Addresses the question of Fatalini and Schindler about the algebraic independence of reals in mutually generic forcing extensions. Proves that the transcendence degree of the reals in V[G_0,...,G_{n-1}] over the field generated by the reals from any proper subcollection is maximal, i.e., continuum. The proof combines De Bondt's really-closure trick (via a countable family of C^1 closures) with a novel non-similarity argument on perfect rectangles to prevent representation of new reals from intermediate models, first in the n=2 case and then in general n. The results extend known Cohen-forcing cases and contribute to the broader understanding of the algebraic structure of forcing extensions, with potential further explicit witness constructions.

Abstract

Let ,..., be mutually generic over , each adding at least one new real over . We show that the transcendence degree of the reals of is maximal (of size continuum) over the field generated by reals coming from models , for a proper subset of . This answers a question of Fatalini and Schindler.

Paper Structure

This paper contains 4 sections, 11 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two generics
  4. Any number of generics

Key Result

Theorem 1.1

Let $\mathbb{P}_0$, …, $\mathbb{P}_{n-1}$ be forcing notions adding new reals and let $G = G_0 \times \dots \times G_{n-1}$ be $\mathbb{P}_0 \times \dots \times \mathbb{P}_{n-1}$-generic over $V$. Then, in $V[G]$, $\mathbb{R}$ has transcendence degree $\mathfrak{c}$ over the field generated by $\big

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2: De Bondt, see Fatalini2025
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 13 more