Paradoxes of the infinite -- how to make two balls out of one (Q2875248)

This article is a written and revised form of a lecture the author gave on the occasion of the 50th jubilee of the Gauß Society. In the introduction, he gives a statement on infinity which Carl Friedrich Gauß wrote to his disciple Heinrich Christian Schumacher in 1831:NEWLINENEWLINE\textit{``\dots so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine façon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen gestattet ist \dots''}NEWLINENEWLINEEnglish translation (taken from Wikipedia):NEWLINENEWLINE\textit{``\dots [I] protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction \dots''}NEWLINENEWLINEBut Gauß was wrong in this respect.NEWLINENEWLINEIn the following chapters of his article the author follows the history of the subject ``infinity'' in mathematics. First he quotes Galileo Galilei's famous proof that there are just as many natural numbers as even numbers. This and other strange things can be illustrated by the well-known ``Hilbert's Hotel''. Furthermore, the author shows that natural numbers and negative numbers are also countable, just as rational numbers. Real numbers must be of another kind of infinity, because they are not countable. The author uses the common indirect proof introduced by Georg Cantor. In the next chapters he passes over to two- and three-dimensional sets. In the 20th century Henri Lebesgue developed a theory of measures and it became possible to construct and analyse very strange sets in Euclidian space. In 1914, \textit{Felix Hausdorff } in [Math. Ann. 75, 428--433 (1915; JFM 45.0128.05)] discovered a way to dissect the surface of a sphere into two parts, each an exact copy of the given one. In 1924, \textit{Stefan Banach} and \textit{Alfred Tarski} extended this paradoxical result to the whole ball [Fundam. Math. 6, 244--277 (1924; JFM 50.0370.02)].NEWLINENEWLINEIn his paper, the author follows the books of \textit{S. Wagon} [The Banach-Tarski paradox. Cambridge etc.: Cambridge University Press (1985; Zbl 0569.43001)] and \textit{L. M. Wapner} [Aus 1 mach 2. Wie Mathematiker Kugeln verdoppeln. Transl. from the American by Harald Höfner and Brigitte Post. Heidelberg: Spektrum Akademischer Verlag (2008; Zbl 1151.00008)]. He does not develop new ideas, but he gives a clearly arranged outline of the subject ``infinity'' in mathematics.