The Hausdorff paradox is a paradox in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

named after

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...

. It involves the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

(a 3-dimensional sphere in

). It states that if a certain

countable In mathematics, a set is countable if either it is 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 from it into the natural numbers ...

subset is removed from

, then the remainder can be divided into three disjoint subsets

and

such that

and

are all congruent. 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 ''Grundzüge der Mengenlehre'' (German for "Basics of Set Theory") is a book on set theory written by Felix Hausdorff. First published in April 1914, ''Grundzüge der Mengenlehre'' was the first comprehensive introduction to set theory. Besides t ...

'', 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 be p ...

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

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

. 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 origina ...

"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 mathematical discipline of measure theory, a Banach measure is a certain type of content used to formalize geometric area in problems vulnerable to the axiom of choice. Traditionally, intuitive notions of area are formalized as a class ...

, 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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...

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 analysis Measure theory

See also

References

Further reading

External links