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140 Appendix at Hilbert’s reaction to the (set-theoretic) paradoxes which were unsettling the
foundations of mathematics at the turn of the 19th to the 20th century.
A.2 Hilbert and the paradoxes
It is well-known that the discovery of an antinomy in Frege’s Grundgeset- zen der Arithmetik [Fre03] by Russell gave rise to a profound discussion of the paradoxes in the mathematical context. For a historical presentation of the discussion of the paradoxes in the literature, a debate involving Russell, Peano, Henri Poincare´ (1854–1912) and others, see e.g., [Gar92]. However, Hilbert’s role in this discussion seems to be underestimated, probably due to the simple fact that he essentially did not publish on this topic.27 (As an ex- ception one might consider his contribution to the 3rd International Congress of Mathematicians in Heidelberg [Hil05c] which also contained the first sketch of what later became “Hilbert’s programme”.) But, as we know from his corre- spondence with Georg Cantor (1845–1918) [Can91, p. 388–390] and Frege [Fre76, letter XV/9], he was involved in this matter from the very beginning. In particular, a paradox found by him has to be considered one of the main motivations for his interest in the foundations, an interest which later resulted in the introduction of proof theory as a new discipline in mathematical logic and—therein—“Hilbert’s programme”, the research programme searching for a consistencyproofofformaltheoriesbyfinitisticmeans.28 Hisparadox,whichis closely related to Cantor’s paradox, is discussed in detail in [PK02]. Kanamori has put it into the following very compact, modern form (as a non-existence proof rather than a paradox), [Kan04, S. 490]:29
There is no set S satisfying (a) if X ∈ S, then its power set P(X)∈S,and(b)ifT ⊆S,thenitsunionT ∈S. Supposethat there were such an S. Then P(S) ∈ S by (b) and then (a). But then, P( S) ⊆ S, which is a contradiction!
Hilbert was constructing such a set S by use of two set formation prin- ciples, namely the Additionsprincip (addition principle) and Belegungsprincip (mapping principle), “nach aller bisherigen Mathematik und Logik unbedenk- lichen Principen” [Hil05a, p. 206]30. Therefore, he reaches a contradiction,
27Hilbert himself comments on this fact with respect to his own paradox (see below): “Ich habe diesen Widerspruch nicht publiciert; er ist aber den Mengentheoretikern, insbesondere G. Cantor, bekannt.” [Hil05a, S. 204] (“I didn’t publish this contradiction; but it is known to the set theorists, in particular to G. Cantor”).
28For the history of Hilbert’s programme(s), cf. [Sie99].
29An implementation of Hilbert’s paradox in a proof system called Goedel, an extension of Mathematica, was given by Johan G. F. Belinfante and Benjamin Lamothe and can be found at: http://www.math.gatech.edu/~belinfan/research/autoreas/goedel/nb/cowork/ hilbert.nb.
30“principles unobjectionable according to all previous mathematics and logic” [PK02, p. 168].