Page 164 - Textos de Matemática Vol. 40

152 Appendix
However, Hilbert’s argumentation can be considered as an elementary undefinability result (rather than as a contribution to the debate on the para- doxes). In this perspective, it contains the structure of modern undecidability results, as we have shown above. We do not claim that Hilbert anticipated these results (which are even contrary to his fundamental philosophical attitude with respect to mathematical knowledge), and the undecidability results were found independent of Hilbert’s paradoxical function. But the similarity of the arguments deserves attention.
With respect to another point—the use of self-application in the definition of the paradoxical function—we tried to show that Hilbert might have influ- enced the later introduction of this concept in mathematics by Scho¨nfinkel, Curry and Church and even its implicit occurrence in von Neumann’s com- puter model. Except for the mentioned letter of Curry to Hilbert [CF58, p. 185] which contained a discussion of Russell’s paradox in combinatory logic, presumably by use of self-application, there is no indication that Hilbert him- self showed further interest in the concept of self-application. However, it can be stated that the history of self-application already started more than 100 years ago, about 15 years before the work of Scho¨nfinkel, when Hilbert used it for the definition of his paradoxical function.