Banach-Tarski paradox

A sphere can be broken down into a finite number of parts, from which two spheres the size of the original can be put together.

The Banach-Tarski paradox or also the Banach and Tarski theorem is a mathematical statement that demonstrates that the notion of volume cannot be generalized to arbitrary sets of points . Then you can divide a sphere into three or more dimensions in such a way that its parts can be joined together again to form two seamless spheres, each of which has the same diameter as the original one. The volume doubles without clearly showing how volume should be created out of nothing through this process. This paradox demonstrates that the mathematical model of space as a point set in mathematics has aspects that cannot be found in physical reality.

Explanation

The paradox is explained formally mathematically by the fact that the spherical parts are so complex that their volume can no longer be defined in a suitable sense. More precisely: It is impossible to define a content that is invariant in motion on the set of all subsets of three-dimensional space , which assigns a volume unequal to zero or infinite to spheres. A content is a function that assigns a positive real number or infinity to each set from a given range of sets , called the volume of the set, so that in particular the volume of the union of two non-overlapping sets is equal to the sum of the volumes of the individual sets. A content is motion-invariant if the volume of a set does not change when it is rotated, displaced or mirrored. Every mathematical concept of volume that is supposed to be a movement-invariant content or even a movement-invariant measure must therefore be restricted in such a way that it is not defined for certain sets, such as these sets into which the sphere can be broken down. The volume is then only defined for sets that lie in Borel's σ-algebra or are Lebesgue-measurable . These quantities do not count towards this. In a certain sense they are infinitely filigree and porous or like a cloud of dust. The mathematical existence of such sets is not self-evident: To prove the existence of non-measurable subsets in - dimensional , real space one needs the axiom of choice or weaker forms of it, which cannot be derived from the Zermelo-Fraenkel set theory . Measurable point sets, on the other hand, behave additively with regard to their volume. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle d}$ ${\displaystyle \mathbb {R} ^{d}}$

The Polish mathematicians Stefan Banach and Alfred Tarski carried out a mathematical proof of existence in 1924 and showed that in the case of the sphere, breaking it down into only six parts is sufficient. On the other hand, it is impossible to provide constructive proof in the sense of an instruction how a sphere is actually to be cut into six parts in order to be able to assemble them into two spheres of the same volume.

General formulation

In a more general formulation of this theorem, the starting and ending bodies can differ by any volume factor and, apart from certain restrictions, can also have any different shape. The general formulation of this mathematical theorem in spaces with three and more dimensions is:

Let be an integer and be bounded sets with non-empty interiors . Then there is a natural number and a disjoint decomposition of and associated motions such that is the disjoint union of the sets . ${\displaystyle d\geq 3}$ ${\displaystyle X,Y\subset \mathbb {R} ^{d}}$ ${\displaystyle n}$ ${\displaystyle X_{1},\dots ,X_{n}}$ ${\displaystyle X}$ ${\displaystyle \beta _{1},\dots ,\beta _{n}}$ ${\displaystyle Y}$ ${\displaystyle \beta _{1}(X_{1}),\dots ,\beta _{n}(X_{n})}$

Evidence sketch

The core of the proof is based on group theory, since rotations in space can be mathematically described as elements of a group that are linked to one another and can operate on other objects:

The combination of several rotations is in turn a rotation.

The link operation is associative.

In a link chain, the neighboring elements are eliminated and because they are inverse to one another. This also applies to rotations. ${\displaystyle g_{1}\dots g_{i}g_{i}^{-1}\dots g_{n}}$ ${\displaystyle g_{i}}$ ${\displaystyle g_{i}^{-1}}$

Rotations operate on points or sets of points in space by changing their position.

A subset of a group creates a subset by linking the elements of the subset and their inverses in all possible combinations. If the subset is finite, such a subgroup is said to be finitely generated. The rotation by 120 ° ( ), for example, creates a subgroup of three elements and is isomorphic to the cyclic group . In principle, it is possible that the elements of subgroups created in this way have several representations with regard to the link between the producers. For the proof of the Banach-Tarski paradox, however, a group is needed in which this is not the case. ${\displaystyle \phi }$ ${\displaystyle \{1,\phi ,\phi ^{2}\}}$ ${\displaystyle C_{3}}$

The free group with its generators and consists abstractly defined of words above the alphabet , in which no inverses are adjacent. The connection of the group represents the concatenation, whereby any inverse pairs that may have arisen are iteratively removed until such pairs no longer appear in the connected word. The empty word represents the neutral element of the group. The representation of each element of as a shortened word is clear. ${\displaystyle F_{2}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle E:=\{a,a^{-1},b,b^{-1}\}}$ ${\displaystyle \varepsilon }$ ${\displaystyle F_{2}}$

Groups that have (finally) been created can be visualized using a Cayley graph. The corners of the graph are elements of the group, the edges are associated with a generator. In the Cayley graph there is a directed edge labeled from corner to corner if and only if a generator is and applies. The Cayley graph of is a triangle, so it is finite and has a circle, while the Cayley graph of has an infinite number of vertices and edges, but no circles. ${\displaystyle g}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle g}$ ${\displaystyle xg=y}$ ${\displaystyle C_{3}}$ ${\displaystyle F_{2}}$

We now define the sets and . ${\displaystyle S(a),S(a^{-1}),S(b)}$ ${\displaystyle S(b^{-1})}$

${\displaystyle S(a)}$let out the set of all words that begin with on the left , analogously to the other S-sets. It is important to realize that for every word , the representation is abbreviated as required above. ${\displaystyle F_{2}}$ ${\displaystyle a}$ ${\displaystyle ag_{1}\dots \in S(a),g_{1}\neq a^{-1}}$

The sets S ( a ), S ( a ⁻¹ ) and aS ( a ⁻¹ ) in the Cayley graph of F ₂

The following disjoint decomposition applies:

${\displaystyle F_{2}=\{e\}\cup S(a)\cup S(a^{-1})\cup S(b)\cup S(b^{-1})}$

But it also applies

${\displaystyle F_{2}=aS(a^{-1})\cup S(a),\,}$

and

${\displaystyle F_{2}=bS(b^{-1})\cup S(b)\,}$,

there

${\displaystyle {\begin{aligned}aS(a^{-1})&:=\{ax|x\in S(a^{-1})\}=\{aa^{-1}\}\cup \{ax|x=a^{-1}g_{0}\dots ,a\neq g_{0}\in E\}\\&=\{e\}\cup \{aa^{-1}g_{0}\dots |a\neq g_{0}\in E\}=\{e\}\cup S(b)\cup S(b^{-1})\cup S(a^{-1})\end{aligned}}}$

and

${\displaystyle {\begin{aligned}bS(b^{-1})&:=\{bx|x\in S(b^{-1})\}=\{bb^{-1}\}\cup \{bx|x=b^{-1}g_{0}\dots ,b\neq g_{0}\in E\}\\&=\{e\}\cup \{bb^{-1}g_{0}\dots |b\neq g_{0}\in E\}=\{e\}\cup S(a)\cup S(b^{-1})\cup S(a^{-1}).\end{aligned}}}$

This paradoxical decomposition of is essential for the later doubling of the sphere. ${\displaystyle F_{2}}$

But first we need to find rotations that behave like the free group . Rotation groups like these are not suitable, and generators that represent a rotation around the angle are also unsuitable , because if they do, the group created has at most elements. Let's turn around an irrational fraction of , e.g. B. with , and be a corresponding rotation around the x-axis, as well as one around the y-axis, then it can be shown that the group created is isomorphic . ${\displaystyle F_{2}}$ ${\displaystyle C_{3}}$ ${\displaystyle q\pi ,q\in \mathbb {Q} }$ ${\displaystyle q={\tfrac {k}{n}}}$ ${\displaystyle 2n}$ ${\displaystyle \pi }$ ${\displaystyle \theta =\arccos \left({\tfrac {1}{3}}\right)}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle F_{2}}$

Group elements can operate on objects and sets of objects; here rotations are supposed to do this on points in space. Be a crowd on which a group operates. Then the operation is a function and, for a fixed one, the path from below . Transferred to the rotations is the set of all points that can be reached from the starting point via all conceivable rotations . Although the Cayley graph describes the relationships among group elements, it helps to visualize the path of an object. If you consider the intersection of the neutral element as an object on which it operates, then the set of all intersections in the Cayley graph is, so to speak, the path of (under certain conditions, see below). ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle M\to M:m\mapsto gm}$ ${\displaystyle Gm:=\{gm|g\in G\}}$ ${\displaystyle m\in M}$ ${\displaystyle m}$ ${\displaystyle G}$ ${\displaystyle F_{2}m}$ ${\displaystyle m}$ ${\displaystyle F_{2}}$ ${\displaystyle e}$ ${\displaystyle m}$ ${\displaystyle F_{2}}$ ${\displaystyle m}$

Let H be the isomorphic group of rotation that is on the unit sphere , i.e. H. the surface of the unit sphere operates. The set of all orbits from H on is a partition of . All orbits are pairwise disjoint and their union yields itself. The axiom of choice allows us to choose a representative from each orbit, hence be the set of these representatives. Be further ${\displaystyle F_{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle M}$

${\displaystyle S(a)M=\{gm|g\in S(a),m\in M\}}$, analog for ${\displaystyle S(b),\dots }$

${\displaystyle B=a^{-1}M\cup a^{-2}M\cup \dots }$

${\displaystyle A_{1}=S(a)M\cup M\cup B}$

${\displaystyle A_{2}=S(a^{-1})M\setminus B}$

${\displaystyle \displaystyle A_{3}=S(b)M}$

${\displaystyle \displaystyle A_{4}=S(b^{-1})M}$

The sets to are disjointed in pairs and form the complete sphere (except for a zero set) . The following applies: ${\displaystyle A_{1}}$ ${\displaystyle A_{4}}$ ${\displaystyle S^{2}}$

${\displaystyle \displaystyle aA_{2}=A_{2}\cup A_{3}\cup A_{4}}$

${\displaystyle \displaystyle bA_{4}=A_{1}\cup A_{2}\cup A_{4}}$

The tripling of the quantities and results from the property of the previously defined S-quantities, which is also reflected in the Cayley graph. If the red elements are "rotated to the right", the amount of blue elements is obtained from . ${\displaystyle A_{2}}$ ${\displaystyle A_{4}}$ ${\displaystyle S(a^{-1})}$ ${\displaystyle aS(a^{-1})}$

Finally we connect every point on the sphere with a ray to the origin of the unit sphere. The decomposition of the sphere induces a decomposition of the sphere (down to the origin), which can be rotated to two spheres of the volume of the origin sphere. ${\displaystyle S^{2}}$

Other aspects

The evidence sketch above has hidden some aspects that need to be considered for a completed evidence. The set of cross points in the Cayley graph is only the trajectory of the center point below if the operation has no fixed points, i.e. H. if for all . For rotations on , this is almost correct, except for the two poles of the rotation. However, since H is countable and every H has exactly two fixed points, only a countable number of points are not rotated when rotating. It can be shown, however, that if (for something countable ) has a paradoxical decomposition, it also exists for one (Hausdorff paradox). ${\displaystyle m}$ ${\displaystyle F_{2}}$ ${\displaystyle gm\neq m}$ ${\displaystyle g\in F_{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle g\in }$ ${\displaystyle S^{2}}$ ${\displaystyle S^{2}\setminus D}$ ${\displaystyle D}$ ${\displaystyle S^{2}}$

Situation in one and in two dimensions

This theorem is not valid in the plane and on the straight line. There is movement-invariant content on the set of all subsets, which assign circles or lines to their usual areas or lengths. However, these hardly play a role in mathematics, because on the one hand they are not clearly defined by the areas of circles or lengths of lines, and they are not dimensions, i.e. H. the union of countably many sets that do not overlap may have a different content than the sum (in the sense of a series ) of the individual contents. However, this property of dimensions is required in very many situations, which is why one is usually content with content in the one- and two-dimensional, which are only defined on certain subsets, but are even dimensions. The non-existence of a motion-invariant measure on the set of all subsets of the straight line or plane is shown (using the axiom of choice) by Vitali's theorem with the existence of the so-called Vitali sets .

In 1990 Miklós Laczkovich was able to show that for some surfaces at least one sentence similar to the above applies, but without the “paradox” of a change in volume. According to this, two surfaces of the same size with a sufficiently smooth edge also have the same decomposition. In this sense, it is possible, for example, to square the circle , albeit not with a compass and ruler. The number of parts required for a constructive solution was estimated by Laczkovich at about , although the sizes of the larger parts were not clearly defined according to Laczkovich. ${\displaystyle 10^{50}}$

However, the theorem cannot be proven without some form of axiom of choice. In 1970 Robert M. Solovay was able to show, assuming the existence of an unreachable cardinal number , that a model of the Zermelo-Fraenkel set theory exists in which all sets are Lebesgue measurable . It is even possible to maintain the validity of a weakened version of the axiom of choice, namely the axiom of dependent choice (DC), which is sufficient for many proofs of elementary analysis. In addition, it could be achieved in this model that every subset of the real numbers has the Baire property and that every uncountable subset of the real numbers contains a non-empty perfect subset . These two statements also contradict the general axiom of choice.

literature

• Leonard M. Wapner: Make 2 out of 1 . How mathematicians double balls. Spektrum Akademischer Verlag, Heidelberg 2007, ISBN 978-3-8274-1851-7 ( Spectrum article on the book [accessed on May 5, 2012] American English: The Pea and the Sun. Translated by Harald Höfner and Brigitte Post, demanding , but for non-mathematicians understandable presentation of the Banach-Tarski theorem including the necessary set-theoretical basics).
• Stefan Banach, Alfred Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes . In: Fundamenta Mathematicae . tape 6 , 1924, ISSN  0016-2736 , pp. 244–277 ( full text [PDF; accessed on May 5, 2012]).

Web links

• Reinhard Winkler: How do you make 2 out of 1? - Derivation with the means of school mathematics, in html and pdf versions.
• The spherical miracle - the paradox of Banach-Tarski on the math planet u. A. with a construction of the required decomposition of the sphere.
• Thomas Neukirchner The Banach-Tarski Paradox , PDF file
• Vsauce.The Banach – Tarski Paradox. Popular science explanations in English . on YouTube

Individual evidence

1. ↑ Jürgen Elstrodt : Measure and integration theory . Springer , Berlin, Heidelberg 1996, ISBN 3-540-15307-1 , pp. 4 .