On a paradox of Hilbert and Bernays
type: Item
instance of: scholarly article
title: On a paradox of Hilbert and Bernays
zbMATH Open document ID: 0869.03005
DOI: 10.1023/A:1017900703234
author: Graham Priest
published in: Journal of Philosophical Logic
publication date: 1997-09-02T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03B65
Mathematics Subject Classification ID: 03F30
zbMATH DE Number: 992037
zbMATH Keywords: paraconsistency
zbMATH Keywords: denoting
zbMATH Keywords: Hilbert-Bernays paradox
zbMATH Keywords: natural language
zbMATH Keywords: denotation function
Wikidata QID: Q56852436
reviewed by: Jim Mackenzie
MaRDI profile type: Publication
cites work: Q4342084
cites work: Q4052072
cites work: Q5772177
cites work: Q5812175
cites work: Q3852153
cites work: Q3992933
cites work: The Trivial Object and the Non-Uiviality of a Semantically Closed Theory with Descriptions
DBLP publication ID: journals/jphil/Priest97
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description: scientific article; zbMATH DE number 992037