On a paradox of Hilbert and Bernays
\textit{D. Hilbert} and \textit{P. Bernays}, in their Grundlagen der Mathematik, Vol. II (1939; Zbl 0020.19301), proved that, in a language for arithmetic which can refer to its own terms, any functional term, e.g. the successor function, of arithmetic has a fixed point. Since no number is its own successor, they concluded that the denotation function of arithmetic cannot be represented in arithmetic. Even a paraconsistentist cannot accept this result that every functional term has a successor. But if we apply the argument to natural language, we cannot but accept that ``is the denotation of'' in English expresses the denotation function for English, and hence we cannot accept the result as a reductio of that assumption. (Cf. the parallel case with Tarski's theorem applied to natural languages, with ``is true'' in place of ``denotes''.) The present paper argues that the most plausible conclusion is that not all terms denote (invalidating some forms of Meinongianism). This is compared with an analogous result (taking Gödel codes as names, and decoding as a denotation function) in recursion theory, where \textit{S. C. Kleene}, Introduction to metamathematics (1952; Zbl 0047.00703), ch. 12, moved to a theory of partial functions, and Kleene's ``strong identity'', \(\simeq\), provides the usual trade-off between expressive completeness and inconsistency.
scientific article; zbMATH DE number 992037
scientific article