The Hahn-Banach theorem implies the Banach-Tarski paradox

The Banach-tarski Paradox

Stefan Banach and Alfred Tarski introduced the phrase: “a pea can be chopped up and reassembled into the Sun,” a seemingly impossible concept. Using this theorem as motivation, this paper will explore the existence of non-measurable sets and paradoxical decompositions as well as provide a sketch of the proof of the paradox.

The Banach-Tarski Paradox

Author’s note: This paper was originally written for my Minor Thesis requirement of the Ph.D. program at Harvard University. The object of this requirement is to learn about a body of work that is generally not in one’s own field, in a short period of time (3 weeks), and write an article about it. I chose this topic because I thought it would be quite interesting to learn, and even more fun to ...

The Banach-tarski Paradox

The Banach-Tarski paradox is one of the most celebrated paradoxes in mathematics. It states that given any two subsets A and B of R, which are bounded and have non-empty interior, it is possible to ‘cut’ A into a finite number of pieces which can be moved by rigid motions (translations and rotations) to form exactly B. This has many amusing consequences; it is most commonly stated as the assert...

The Banach-Tarski Paradox

What is a set? In the late 19th century, Georg Cantor was the first to formally investigate this question, thus founding the study of set theory as a mathematical discipline. Cantor’s definition of a set consisted of two intuitive principles. The first, Extensionality, asserts that two sets are equal exactly when they have the same members; the second, Comprehension, asserts that from any condi...

The Banach-Tarski Paradox