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I want to learn Banach Tarski's Paradox but I only know Algebra (little really), so I decided to spend some time on this but I don't know where to start.

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  • $\begingroup$ You don't need THAT much math, other than set theory, but honestly, it's hard to understand the intuition behind what the result means without a stronger mathematical foundation. $\endgroup$ – Don Thousand Feb 29 at 16:44
  • $\begingroup$ Group theory is probably helpful (you don't need too strong an understanding of group theory: a passing knowledge of what a free group is should suffice). $\endgroup$ – Don Thousand Feb 29 at 16:47
  • $\begingroup$ You might find it helpful to begin with The Pea and the Sun. A Mathematical Paradox by Leonard M. Wapner (2005), and then ask more specific background questions as you read through Wapner's book. By "more specific", I mean that in addition to specifying a particular topic (cardinality of sets, group theory, rotation matrices, etc.), by providing more specific information about what you do know and what you don't know about that topic. $\endgroup$ – Dave L. Renfro Feb 29 at 17:09
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I don't think you need too much in terms of prerequisites.

An excellent reference is

MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-107-04259-9.

The book is essentially self-contained, and it may make more sense (be more efficient) to read it and look at other sources as the need appears. You need some group theory, as the paradox depends on properties of free groups, but what is needed is explained in the book. You also need the axiom of choice, but the way it is used is fairly direct. I suppose one needs a little about the basics of measure theory, to understand why this is a "paradox", but this is also introduced in the book. (Deeper knowledge of these and other topics is required as the book goes on, but for the basics you can essentially just delve in.)

A nice feature of the book is that, while remaining user-friendly, it puts the paradox in context, relating it to other topics, some more advanced or even active areas of research (amenability, for instance). Another is that it is basically up-to-date. I know of only one (technical, but really neat) result that was proved after this edition came out (the solution by Marks and Unger of one of the open problems listed in the book: a constructive proof of Laczkovich's result).

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