Let be the unit sphere in the Euclidean space . Then it is possible to take “half” and “a third” of such that both of these parts are essentially congruent (we give a formal version in a minute). This sounds paradoxical: wouldn’t that mean that half of the sphere’s area is equal to only a third? The “paradox” resolves itself if one takes into account that one can choose non-measurable subsets of the sphere which ostensively are “half” and a “third” of it, using geometric congruence as means of comparison.
Let us now formally state the Theorem.
Theorem (Hausdorff paradox [H]).
A crucial ingredient to the proof is the http://planetmath.org/node/310axiom of choice, so the sets , and are not constructible. The theorem itself is a crucial ingredient to the proof of the so-called Banach-Tarski paradox.
- H F. Hausdorff, Bemerkung über den Inhalt von Punktmengen, Math. Ann. 75, 428–433, (1915), http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28919http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28919 (in German).
|Date of creation||2013-03-22 15:16:12|
|Last modified on||2013-03-22 15:16:12|
|Last modified by||GrafZahl (9234)|