First stated in , the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form. The Banach-Tarski paradox is a theorem in geometry and set theory which states that a. THE BANACH-TARSKI PARADOX. ALLISON WU. Abstract. Stefan Banach and Alfred Tarski introduced the phrase: “a pea can be chopped up.
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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Banach-Tarski paradoxcantor’s theoremnon-measurabilityparadosset theory by Terence Tao 42 comments. The standard modern foundation of mathematics is constructed using set theory. In a pure set theory, the primitive objects would themselves be sets as well; this is useful for studying the foundations of mathematics, but for most mathematical purposes it is more convenient, and less conceptually confusing, to refrain banadh modeling primitive objects as sets.
After all, if one deals solely with finite sets, then there is no need to distinguish between countable and uncountable infinities, and Banach-Tarski type paradoxes cannot occur. On the other hand, many statements in infinitary mathematics can be reformulated into equivalent statements in finitary mathematics involving only finitely many points or numbers, etc. So, one may ask: Though this is not quite the end of the story; after all, one also has for every natural numberor equivalently that the union of a finite set and an additional element cannot be enumerated by itself, but the former statement extends to the infinite case, while the latter one does not.
What causes these two outcomes to be distinct?
On the other hand, it is less obvious what tarwki finitary version of the Banach-Tarski paradox is. Note that this paradox is available only in three and higher dimensions, but not in one or two dimensions; so presumably a finitary analogue of this paradox should paradx make the same distinction between low and high dimensions. It seems that the easiest way to accomplish this is to avoid the use of set theory, and replace sets by some other concept. Taking inspiration from theoretical computer science, I decided to replace concepts such as functions and sets by the concepts of algorithms and oracles instead, with various constructions in set theory being replaced instead by computer language pseudocode.
Finally one can allow constructions indexed by arbitrary ordinals i. I should caution that this is a conceptual exercise rather than a rigorous one; I have not attempted to formalise these notions to the same extent that set theory is formalised.
Thus, for instance, I have no explicit system of axioms that algorithms and oracles are supposed to obey. Of course, these formal issues have been explored in great depth by logicians over the past century or so, but I do not wish to focus on these topics in this post.
A second caveat is that the actual semantic content of this post is going to be extremely low.
Nevertheless I believe this viewpoint is somewhat clarifying as to the nature of these paradoxes, and as to how they are not as fundamentally tied to the nature of sets tarsii the nature of infinity as one might first expect.
In this post only, I will colour a statement red if it assumes the axiom of choice. For the rest of the course, the axiom of choice will be implicitly assumed throughout.
Banach-Tarski paradox | What’s new
The famous Banach-Tarski paradox asserts that one can take the unit ball in three dimensions, divide it up into finitely many pieces, and then translate and rotate each piece so that their union is now two disjoint unit balls.
As a consequence of this paradox, it is not possible to create a finitely additive measure on that is both translation and rotation invariant, which can measure every subset ofand which gives the unit ball a non-zero measure.
This paradox helps explain why Lebesgue measure which is countably additive and both translation and rotation invariant, and gives the unit ball a non-zero measure cannot measure every set, instead being restricted to measuring sets that are Lebesgue measurable. On the other hand, it is not possible to replicate the Banach-Tarski paradox in one or two dimensions; the unit interval in or unit disk in cannot be rearranged into two unit intervals or two unit disks using only finitely many pieces, translations, and rotations, and indeed there do exist non-trivial finitely additive measures on these spaces.
However, it is possible to obtain a Banach-Tarski type paradox in one or two dimensions using countably many such pieces; this rules out the possibility of extending Lebesgue measure to a countably additive translation invariant measure on all subsets of or any higher-dimensional space. In these notes I would like to establish all of the above results, and tie them in with some important concepts and tools in modern group theory, most notably amenability and the ping-pong lemma.
This material is not required for the rest of the course, but nevertheless has some independent interest.
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