The Hausdorff paradox is a paradox in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

named after

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (''à mogré' (Fr.) = "according to my taste"), who is considered to be one of the founders of modern topology and who contributed sig ...

. It involves the

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

(the surface of a 3-dimensional ball in

). It states that if a certain

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

subset is removed from

, then the remainder can be divided into three disjoint subsets

and

such that

and

are all

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

. In particular, it follows that on

S^2

there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of

is simultaneously

1/3

1/2

, and

2/3

of the non-zero measure of the whole sphere). The paradox was published in ''

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' in 1914 and also in Hausdorff's book, '' Grundzüge der Mengenlehre'', the same year. The proof of the much more famous

Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then ...

uses Hausdorff's ideas. The proof of this paradox relies on the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

. This paradox shows that there is no finitely additive measure on a sphere defined on ''all'' subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no ''countably'' additive measure defined on all subsets.) The structure of the group of rotations on the sphere plays a crucial role here the statement is not true on the plane or the line. In fact, as was later shown by Banach,

Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...

"Sur le problème de la mesure"
Fundamenta Mathematicae 4: pp. 7–33, 1923; Banach
"Sur la décomposition des ensembles de points en parties respectivement congruentes"
Theorem 16, Fundamenta Mathematicae 6: pp. 244–277, 1924. it is possible to define an "area" for ''all'' bounded subsets in the Euclidean plane (as well as "length" on the real line) in such a way that congruent sets will have equal "area". (This

Banach measure In the mathematics, mathematical discipline of measure theory, a Banach measure is a certain way to assign a size (or area) to all subsets of the Euclidean plane, consistent with but extending the commonly used Lebesgue measure. While there are c ...

, however, is only finitely additive, so it is not a measure in the full sense, but it equals the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal area. __NOTOC__

References

External links

Hausdorff Paradox
on ProofWiki Mathematical paradoxes Theorems in mathematical analysis Measure theory

See also

References

Further reading

External links