Hausdorff paradox
Encyclopedia
In mathematics
, the Hausdorff paradox, named after Felix Hausdorff
, states that if you remove a certain countable
subset of the sphere
S2, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent
. In particular, it follows that on S2 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 A is both 1/3 and 1/2 of the non-zero measure of the whole sphere).
The paradox was published in Mathematische Annalen
in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre
, the same year. The proof of the much more famous Banach–Tarski paradox
uses Hausdorff's ideas.
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, it is possible to define an "area" for all bounded subsets in the Euclidean plane (as well as "length" on the real line) such that congruent sets will have equal "area". (This Banach measure
, however, is only finitely additive, so it is not a measure
in the full sense, but it equals the Lebesgue measure
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hausdorff paradox, named after Felix Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish 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, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
, states that if you remove a certain countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
subset of the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
S2, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
. In particular, it follows that on S2 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 A is both 1/3 and 1/2 of the non-zero measure of the whole sphere).
The paradox was published in Mathematische Annalen
Mathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre
Grundzüge der Mengenlehre
Grundzüge der Mengenlehre is an influential book on set theory written by Felix Hausdorff.First published in April 1914, Grundzüge der Mengenlehre was the first comprehensive introduction to set theory...
, the same year. The proof of the much more famous Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
uses Hausdorff's ideas.
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, it is possible to define an "area" for all bounded subsets in the Euclidean plane (as well as "length" on the real line) such that congruent sets will have equal "area". (This Banach measure
Banach measure
In mathematics, Banach measure in measure theory may mean a real-valued function on the algebra of all sets , by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round...
, however, is only finitely additive, so it is not a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
in the full sense, but it equals the Lebesgue measure
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
then they have equal area.