The post Remarks on The Banach-Tarski Paradox first appeared on FifteenEightyFour | Cambridge University Press.

]]>In spite of the above mentioned difficulties, there is a huge interplay between the Banach-Tarski paradox and pure mathematics. The first edition of *The Banach-Tarski Paradox* had a great impact on some difficult problems, including Tarski’s circle-squaring problem and Marczewski’s problem. And it appeared at a time of intensive research in the areas related to the Banach-Tarski theory. We could mention here the solution of Ruziewicz’s problem. The second edition of the “The Banach-Tarski paradox” has similar features. There are several active research areas providing spectacular solutions of some open problems. First of all we mention descriptive combinatorics which provides tools for a solution of the circle-squaring problem with Lebesgue measurable pieces or a radical simplification of the construction of paradoxical decomposition of the unit ball in R3 with the pieces having Property of Baire. This thread leads also to some applications of paradoxical decomposition to the theory of games with incomplete information (which are quite close to physical reality). There is still also an intensive feedback between amenability and paradoxical decomposition. However only recently it appeared that transversality theory and theory of o-minimal structures have some significant meaning for paradoxical decompositions. We hope as in the case of the first edition, the second edition provides a lot of new research.

Find out more about the second edition of *The Banach–Tarski Paradox*, including additional resources

The post Remarks on The Banach-Tarski Paradox first appeared on FifteenEightyFour | Cambridge University Press.

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