| Version 3 |
Version 2 |
| \PMlinkescapeword{constructible} |
\PMlinkescapeword{constructible} |
| \PMlinkescapeword{mean} |
\PMlinkescapeword{mean} |
| \PMlinkescapeword{sounds} |
\PMlinkescapeword{sounds} |
| \PMlinkescapeword{state} |
\PMlinkescapeword{state} |
| \PMlinkescapeword{unit} |
\PMlinkescapeword{unit} |
| Let $S^2$ be the unit sphere in the Euclidean space $\mbb{R}^3$. Then |
Let $S^2$ be the unit sphere in the Euclidean space $\mbb{R}^3$. Then |
| it is possible to take ``half'' and ``a third'' of $S^2$ such that |
it is possible to take ``half'' and ``a third'' of $S^2$ such that |
| both of these parts are essentially congruent (we give a formal |
both of these parts are essentially congruent (we give a formal |
| version in a minute). This sounds paradoxical: |
version in a minute). This sounds paradoxical: |
| wouldn't that mean that half of the sphere's area is equal to only a |
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 |
third? The ``paradox'' resolves itself if one takes into account that |
| one can choose the ``half'' and the ``third'' in a plausible group |
one can choose the ``half'' and the ``third'' in a plausible group |
| theoretical sense, yet the resulting subsets of the sphere are not |
theoretical sense, yet the resulting subsets of the sphere are not |
| measurable. |
measurable. |
|
|
| Let us now formally state the Theorem. |
Let us now formally state the Theorem. |
|
|
| \newtheorem*{thm}{Theorem} |
\newtheorem*{thm}{Theorem} |
| \begin{thm}[Hausdorff paradox~\cite{H}] |
\begin{thm}[Hausdorff paradox~\cite{H}] |
| There exists a disjoint \PMlinkescapetext{decomposition} of the unit sphere $S^2$ in the |
There exists a disjoint \PMlinkescapetext{decomposition} of the unit sphere $S^2$ in the |
| Euclidean space $\mbb{R}^3$ into four subsets $A,B,C,D$, such that the |
Euclidean space $\mbb{R}^3$ into four subsets $A,B,C,D$, such that the |
| following conditions are met: |
following conditions are met: |
| \begin{enumerate} |
\begin{enumerate} |
| \item Any two of the sets $A$, $B$, $C$ and $B\cup C$ are congruent. |
\item Any two of the sets $A$, $B$, $C$ and $B\cup C$ are congruent. |
| \item $D$ is countable. |
\item $D$ is countable. |
| \end{enumerate} |
\end{enumerate} |
| \end{thm} |
\end{thm} |
|
|
| A cruical ingredient to the proof is the axiom of choice, so the |
A cruical ingredient to the proof is the axiom of choice, so the |
| sets $A$, $B$ and $C$ are not constructible. The theorem itself is a |
sets $A$, $B$ and $C$ are not constructible. The theorem itself is a |
|
cruical ingredient to the proof of the so-called Banach-Tarski
|
cruical ingredient to the proof of the so-called \PMlinkid{Banach--Tarski
|
|
paradox.
|
paradox}{4464}.
|
|
|
| \begin{thebibliography}{H} |
\begin{thebibliography}{H} |
|
|
|
\bibitem[H]{H} \textsc{F.~Hausdorff}, Bemerkung über den Inhalt von
|
\bibitem[H]{H} \textsc{F.~Hausdorff}, Bemerkung über den Inhalt von
|
| Punktmengen, \emph{Math.\ Ann.}\ 75, 428--433, (1915). |
Punktmengen, \emph{Math.\ Ann.}\ 75, 428--433, (1915). |
|
|
| \end{thebibliography} |
\end{thebibliography} |
