Page 156 - Textos de Matemática Vol. 40

144 Appendix
And, we can add that g corresponds to Hilbert’s paradoxical function f. Despite this analogy to Russell’s paradox there is no further clue which would relate the paradoxical function with Richard’s paradox or his own one (or Cantor’s). In these paradoxes, a form of self-reference (which is more general than self-application) is only hiddenly present by the use of Cantor’s diagonal- ization method. As we will discuss later, one can consider Hilbert’s paradoxical function as a special case of diagonalization. Therefore, the only relation of all cases discussed by Hilbert is that they involve some form of diagonalization. But there is no indication that Hilbert singled out diagonalization as the com- mon reason for the paradoxes. In fact, in Hilbert’s view this cannot be the reason for the paradoxes since it is a concept which can be used innocently in mathematics. This is exemplified in Cantor’s proof of the uncountability of the real numbers, a proof which Hilbert praises as “einen der sch¨onsten Beweise der Mengenlehre”40 [Hil05a, p. 196]. In contrast, the fact that an operation causes contradictions in some cases, but works fine in other cases, is a prob- lem, in particular, if one can decide only from the result whether the use of an operation is admissible or not. This was emphasized by Hilbert in 1917 in a
discussion of his paradox [Hil17, S. 135]:41
Das eine Mal sollen also zwei mathematische Operationen erlaubt sein, weil sich kein Widerspruch ergibt; das andere Mal aber sollen zwei solche Prozesse unzula¨ssig sein, weil eine Paradoxie folgt, und nur der Erfolg soll entscheiden, was verboten ist und was nicht. [. . . ]
40“one of the most beautiful proofs in set theory” [Moo02, p. 48].
41Hilbert already addressed this point, less drastically, in his contribution to the Interna- tional Congress of Mathematicians [Hil05c] (cited in [Can91, S. 436]):
G. Cantor hat den genannten Widerspruch [den des Begriffs der Gesamtheit aller Dinge] empfunden und diesem Empfinden dadurch Ausdruck verliehen, daß er ”konsistente“ und ”nichtkonsistente“ Mengen unterscheidet. Indem er aber meiner Meinung nach fu¨r diese Unterscheidung kein scharfes Kriterium aufstellt, muß ich seine Auffassung u¨ber diesen Punkt als eine solche bezeichnen, die dem subjektiven Ermessen noch Spielraum l¨aßt und daher keine objektive Sicherheit gew¨ahrt.
(“G. Cantor experienced the mentioned contradiction [of the concept of the totality of all things] and expressed this experience by distinguishing ‘consistent’ and ‘inconsistent’ sets. But, in my opinion, by not establishing a clear criterion for this distinction, I must call his approach in this point one which still leaves latitude for subjective judgement and therefore affords no objective certainty.” See also [Sie02, p. 368].)
In a more general setting, but with the same emphasize on the problem of justification in hindsight, Frege already gave a similiar argument in the introduction to his Grundlagen der Arithmetik [Fre84, p. ix]:
[E]s ist wohl zu beachten, daß die Strenge der Beweisfu¨hrung ein Schein bleibt, mag auch die Schlußkette lu¨ckenlos sein, wenn die Definitionen nur nachtr¨aglich dadurch gerechtfertigt werden, daß man auf keinen Widerspruch gestoßen ist.
(“[I]t must still be borne in mind that the rigour of the proof remains an illusion, however flawless the chain of deduction, so long as the definitions are justified only as an afterthought, by our failing to come across any contradicition” [Fre50, p. ixe].)