Banach-tarski’s paradox — Wikipedia

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Illustration du Paradoxe de Banach-Tarski

In mathematics, and more precisely in geometry, the Paradoxe de Banach-Tarski is a theorem, demonstrated in 1924 by Stefan Banach and Alfred Tarski, who claims that it is possible to cut a ball of usual space

R3{displaystyle {mathbb {R} ^{3}}}
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In a name fine of pieces and to allow these pieces to form two balls identical to the first, to a displacement. This paradoxical result implies that these pieces are non -measurable, otherwise we would obtain a contradiction (the volume being an example of measurement, that means that these pieces have no volume).

The Banach-Tarski paradox is generalized to all

Rn, n ≥ 3 {displaystyle {mathbb {R} ^{n},ngeq 3}}

, but cannot be realized in the plan

R2{displaystyle {mathbb {R} ^{2}}}

.

The demonstration of this result uses the axiom of choice, necessary to build non -measurable sets.

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The group of travel (or isometries direct affins) of

R3{displaystyle {mathbb {R} ^{3}}}

is the whole of all translations and rotations (around an axis) and their compounds, that is to say the whole of all ways of taking a figure in space and moving it or Turn it on itself without distorting it (and in particular without changing its size). A displacement can be seen as a mathematical function g and a figure like a set of points AND . Say that there is a trip g such as g ( AND ) = F , it is simply to say that AND And F have the same shape and the same size, in short, that they are identical to their position.

Two sets are therefore equidable if you can cut the first into pieces and rebuild the second simply by moving the pieces (that is to say by applying a displacement). A set is duplicable if it is equidable in the meeting of two disjointed copies of itself.

Informally, a measure is a mathematical function that satisfies the same conditions as a length. It is therefore a generalization of the length (or volume). A good example of a measurement is the measurement of Lebesgue: if you want to measure an interval, you take its length and if you have a set “in several pieces”, we take the amount of the length of each of the pieces. For example, if two bottles of a liter of wine are placed in two different places, physically there are two distinct objects. This is where the volume shows “its limits”. But mathematically we can consider that these two bottles form only one and the same object whose volume is 2 liters. This is an example of measurement.

More generally, the measurement of a set made up of several disjoint “objects” is the sum of the measures of each objects, the measurement of an “object” without interior is zero, and finally the measurement of an “object” does not Don’t change if you move it. What this paradox asserts is that we can “build” – but using the axiom of choice, therefore in a non -effective way – “twisted” sets so that we cannot Measure, that is to say that one cannot associate them with a value in general (or a particular volume or length) without contradicting the three properties mentioned above. More specifically, if we try to find a way of associating them with a volume, we can prove that by continuing to apply this method, we will find a part that has the same volume as the whole, which is absurd. The paradox claims that you can split a ball from the moment that you pass through a stage where it is cut into non -measurable pieces. Subsequently, these pieces can be reached in a “larger” object without having to say that this object and the starting ball have the same volume since the volume of the result is not the sum of the volumes of the pieces. Also, it is commonly admitted that it is impossible to define the volume of such complex parts.

This is nuanced to say the least by Leroy’s work [ first ] and Simpson [ 2 ] . Indeed, the point of view of local theory makes it possible to identify topological spaces with great finesse. Just as a degree polynomial 2 always has two complex roots counted with multiplicity but that this regularity is hidden if one thinks only of the real roots and without multiplicity, the theory of the locals makes it possible to make the theory more harmonious by “revealing parts space hidden “. In particular, a local can be non-home while containing no point. The theory of the locals makes it possible to define a measurement on all the subcaters (in particular all the parts) of the dimension space d. This seems to contradict the Banach-Tarski paradox (as well as the Vitali paradox), but the contradiction is only apparent. Indeed, the pieces of paradoxical decomposition are disjointed in the sense of sets but not in the sense of the theory of locals: the intersections are sets without point but still non-vides, and which are actually attributed a non-measurement nothing. It is neglecting this hidden mass which leads to the paradox in classical theory. Thus, it does not exist so to define the volume of all parts of the space in a way that respects the naive expectations of what a volume is but there is a satisfactory way of revising this specifications and for which the volume of any sub-local space is well defined.

Either G A group of transformations in a set AND .
Two parts A And B of AND are said equidable (following G ) if there is a finite partition of A , ( A first , …, A n ), and a finite partition of B of the same size, ( B first , …, B n ), as :

For example, any parallelogram is equidable to a rectangle. Equisty is an equivalence relationship: it is symmetrical, reflexive and transitive. Note here that it is not interesting to include homotheties in G . We therefore generally take the group of isometries. We even manage here to limit ourselves to direct isometries (engendered by translations and rotations).

A set C is said to be “declitable” if it is the meeting of two disjoint sets A And B such as A, B And C are equidable.

Demonstrate the Banach-Tarski result is to show that the ball

R3{displaystyle {mathbb {R} ^{3}}}

is duplicable according to the group of travel of

R3{displaystyle {mathbb {R} ^{3}}}

.

This paradox takes place in dimension 3 and more but not in dimension 1 or 2. This is intimately linked to the fact that the group of rotations (vector) of the Euclidean space of dimension D> 2 is not commutative, and more precisely than the Group of isometries affines of this space is an average group if and only if D is less than or equal to 2. The notion of medium -foundability, very fertile, was also introduced by Von Neumann in 1929 to elucidate this paradox.

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