Formalization, syntax and the standard model of arithmetic
The starting point is a discussion between Beth and Carnap from ``Carnap's views on the advantages of constructed systems over natural languages in the philosophy of language'', by \textit{E. W. Beth}, with a reply by Carnap in [P. A. Schlipp (ed.), The philosophy of Rudolf Carnap. Open Court, La Salle (1963)]. Beth distinguishes between \textit{strict usage} of the natural language which requires an intended domain of interpretation and {amplified usage} involving interpretations in any model of the language. If the amplified usage is applied to the syntax and semantics of formal languages, which, according to Bellotti is a necessity of modern mathematics, then one must deal with the problem of nonstandard finiteness brought in by nonstandard models of arithmetic and set theory. Bellotti outlines the Beth-Carnap dispute in detail, and then moves on to discuss views of Skolem and Zermelo on the role of nonstandard models in foundations of mathematics; the status of the standard model of arithmetic (it contains definable nonstandard models, while no nonstandard model carries a definable copy of the standard one); the role of set theory and Tarskian semantics. From the conclusions: ``Since the usual semantics for arithmetic, being set-theoretic, cannot yield the unique model needed for the determinateness of syntax, one cannot isolate any syntactic metatheory with a special status sufficient to preserve it from the instability coming from non-standard interpretability. Nevertheless, as we have seen, we have a notion of formal system \textit{exactly as determinate as we need}, though always inside a certain chosen system, which works as our frame of reference.''
scientific article; zbMATH DE number 5156958