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, and ...
. It involves the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(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
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 mod ...
. In particular, it follows that on
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
,
, and
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 th ...
'', 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 collectio ...
.
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 type of finite measure, content used to formalize geometric area in problems vulnerable to the axiom of choice.
Traditionally, intuitive notions of are ...
, however, is only finitely additive, so it is not a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
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 wit ...
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.
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See also
*
*
References
Further reading
* (Original article; in German)
*{{cite book , first=Felix , last=Hausdorff , title=Grundzüge der Mengenlehre , year=1914 , language=de , url = https://archive.org/details/grundzgedermen00hausuoft/page/n7/mode/2up
External links
Hausdorff Paradoxon ProofWiki
Mathematical paradoxes
Theorems in analysis
Measure theory