A zul F atalini . Paradoxical sets and the Axiom of Choice . Universität Münster 2024. Supervised by Ralf Schindler. MSC: 03E25. Keywords: Axiom of Choice, paradoxical sets, partitions in unit circles, Mazurkiewicz sets, well ordering of the reals.
Year: 2025
Type: article
Abstract: The thesis is thematically divided into two parts: Algebraic closures of certain subfields of the reals (Part I) and Paradoxical sets of reals (Part II). Part I investigates a folklore result about the forcing extension by one Cohen real: the transcendence degree of the reals over the set of reals in the ground model is of cardinality c $\mathfrak {c}$ German c (in the extension). We extend this to the case in which more Cohen reals are added, obtaining the following result: Theorem A. Let X be a finite set of mutually generic Cohen reals over V . In V [ X ] $V[X]$ upper V left bracket upper X right bracket , consider the minimum field F ⊆ R $F\subseteq \mathbb {R}$ upper F subset of or equal to double struck upper R such that F ⊇ ⋃ Y ⊊ X R V [ Y ] $F\supseteq \bigcup _{Y\subsetneq X} \mathbb {R}^{V[Y]}$ upper F superset of or equal to union Underscript upper Y subset of or not equals upper X Endscripts double struck upper R Superscript upper V left bracket upper Y right bracket . Then, in V [ X ] $V[X]$ upper V left bracket upper X right bracket the transcendence degree of R $\mathbb {R}$ double struck upper R over F is continuum. In Part II, we consider some paradoxical sets of reals and study their interaction with the Axiom
Source: Bulletin of Symbolic Logic
Author: Azul Fatalini
Institution: University of Münster
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4416385571
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