FG1 Seminar talk

2018-01-12
Luka Ilić
Hahn-Banach implies Banach-Tarski

Abstract:
Contradicting geometric intuition, the Banach-Tarski Paradox may be regarded as one of the most surprising results of modern mathematics. Its original proof relies on the Axiom of Choice and as such, was heatedly debated throughout the 20th century. More recently (in 1991) a result by Janusz Pawlikowski was published, showing that Choice is not needed in full strength to imply the Banach-Tarski Paradox - he showed that postulating Hahn-Banach suffices. In my talk I will introduce the precise statement of the Banach-Tarski Paradox, present a proof using the Axiom of Choice and additionally present Pawlikowski’s proof, that Hahn-Banach implies Banach-Tarski.