TL;DR
This thesis explores the Banach-Tarski paradox, providing modern proofs, discussing its measure-theoretic significance, and analyzing the role of the axiom of choice in its formulation.
Contribution
It offers updated proofs of the paradox, clarifies historical attributions, and examines the axiomatic foundations necessary for the paradox.
Findings
Modernized proofs of Banach-Tarski paradoxes
Clarification of historical attribution issues
Analysis of axioms needed for the paradoxes
Abstract
This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical significance of the paradox for measure theory is covered, along with its incorrect attribution to Banach and Tarski. Finally, the necessity of the axiom of choice is discussed and contrasted with other axiomatic and topological assumptions that enable the paradoxes.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Mathematical and Theoretical Analysis