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

Azul Fatalini, Ralf Schindler

TL;DR

The paper extends the folklore result that adding a single Cohen real yields maximal transcendence degree of the reals over ground-model reals to the case of a finite set $X$ of mutually generic Cohen reals. It defines the minimal subfield $F$ containing all reals from proper subfamilies and proves that the transcendence degree of $\, ext{R}$ over $F$ in $V[X]$ is the continuum, i.e., $ rak{c}$, by constructing witnesses and deploying a notion of $V$-continuous dependence via an implicit-function-type framework. The authors first treat the case $|X|=2$ and then extend to any finite $|X|$, showing the witness composition of the reals lies outside the relevant subfields and confirming Kanovei and Schindler's question about subfield properness. Overall, the results demonstrate the robustness of maximal algebraic independence under finite Cohen forcing and illuminate the structure of transcendence bases across inner models with multiple added reals.

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.

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

TL;DR

The paper extends the folklore result that adding a single Cohen real yields maximal transcendence degree of the reals over ground-model reals to the case of a finite set of mutually generic Cohen reals. It defines the minimal subfield containing all reals from proper subfamilies and proves that the transcendence degree of over in is the continuum, i.e., , by constructing witnesses and deploying a notion of -continuous dependence via an implicit-function-type framework. The authors first treat the case and then extend to any finite , showing the witness composition of the reals lies outside the relevant subfields and confirming Kanovei and Schindler's question about subfield properness. Overall, the results demonstrate the robustness of maximal algebraic independence under finite Cohen forcing and illuminate the structure of transcendence bases across inner models with multiple added reals.

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.

Paper Structure

This paper contains 5 sections, 16 theorems, 39 equations, 2 figures.

Table of Contents

  1. Prerequisites
  2. Field theory
  3. Real analysis
  4. The transcendence degree of really closed fields
  5. The Main Theorem

Key Result

Theorem 1.3

If $F$ is an extension field of a field $K$ and $X\subseteq F$, then the subfield $K(X)$ consists of all elements of the form where $n\in \omega$, $f,g\in K[x_0, \dots , x_{n-1}]$, $u_0, \dots, u_{n-1}\in X$ and $g(u_0, \dots, u_{n-1})\neq 0$.

Figures (2)

  • Figure 2: Steps $0$ to $(k-1)$ of the construction. The first arrow represents that in Step 0 we take $p_0^0(m)=\mathop{\mathrm{dom}}\nolimits(p_1)$, where $m=\mathop{\mathrm{lh}}\nolimits(p_0)$, and similarly for the other arrows.
  • Figure 3: If $B \subseteq \mathbb{R}^{V[x]}$ is algebraically independent over $\mathbb{R}^V$, then $B$ is also algebraically independent over $\mathbb{R}^{V[y]}$.

Theorems & Definitions (37)

  • Definition 1.2
  • Theorem 1.3: Hungerford2003
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Remark 1.10
  • Definition 1.12
  • ...and 27 more