Page 153 - Textos de Matemática Vol. 40

A note on the history of functional self-application 141
which he rates even more important than the Russell-Zermelo paradox since it is “of purely mathematical nature”(while the latter one is of logical nature), cf. [PK02, p. 163].
In 1905 Hilbert started to present the paradoxes in lectures on the foun- dations of mathematics at the University of G¨ottingen,31 and we can find de- tailed expositions in the unpublished lecture notes of various courses up to 1920, namely: Logische Principien des mathematischen Denkens (Logical principles of the mathematical thinking) in 1905 [Hil05a, Hil05b], Mengenlehre (Set theory) in 1917 [Hil17], and Probleme der mathematischen Logik (Problems of mathe- matical logic) in 1920 [Hil20].32
In the lecture of 1905, the second part is entitled Die logischen Grundla- gen (The logical foundations) and starts with the Chapter Paradoxa der Men- genlehre (Paradoxes of set theory). After some historical and philosophical consideration and a presentation of Cantor’s diagonalization method, Hilbert discusses three contradictions: Richard’s paradox (as it is called nowadays), his own paradox and the Russell/Zermelo paradox33. At the end of the Chapter, Hilbert added a short paragraph on the formal reasons of the contradictions. It is this paragraph which contains the example of functional self-application we are interested in.
A.3 Hilbert’s paradoxical function
We present the German original of this paragraph [Hil05a, S. 213f] followed by an English translation:34
Es wird nun noch nu¨tzlich sein sich klar zu machen, worauf formal jene Widerspru¨che beruhen. Man denke sich ein Ding x, das alle Dinge durchla¨uft. Wir definieren nun eine gewisse Funktion, das ist wieder ein Ding f, und der ”Wert dieser Funktion fu¨r das Argument x“ ist die Combination der beiden Dinge f x. Je nachdem nun die Combination x x von x mit sich selbst 0 ist oder nicht, mo¨ge f x 1 sein oder 0:
fx=0 , wennxx̸=0 fx=1 , wennxx=0;
31In parallel to and partly in interaction with the mathematical discussion of the paradoxes by Hilbert and his students, there was a lively discussion of the paradoxes in the philosophical circle around Leonard Nelson (1882–1927) in G¨ottingen, cf. [Pec90, Pec04].
32For a comprehensive study of these lectures with respect to the paradoxes see [Kah06]. References to these lecture notes can also be found in [Moo02]. The edition of the lecture notes is planned within the series David Hilbert’s Lectures on the Foundations of Mathematics and Physics, 1891–1933 of which at the moment (January 2007) only the first volume (on Geometry) has appeared, [HM04].
33In the lecture notes of 1905 Hilbert is attributing the paradox only to Ernst Zermelo (1871–1953); for the independent discovery of Russell’s paradox by Zermelo, cf. [RT81].
34In the following we call the function in question “Hilbert’s paradoxical function”.