The syntax of nonstandard analysis
type: Item
instance of: scholarly article
title: The syntax of nonstandard analysis
zbMATH Open document ID: 0642.03040
DOI: 10.1016/0168-0072(88)90050-4
author: Edward Nelson
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
review text: The author continuous in this paper his influential study, started with the paper ``Internal set theory: a new approach to nonstandard analysis'' [Bull. Am. Math. Soc. 83, 1165-1198 (1977; Zbl 0373.02040)], of formalizations of principles of nonstandard analysis. While the aim of the first paper was to construct a formal system, Internal Set Theory (IST), which codified a large part of existing mathematical practice of n.a., in the present paper the author shows that ``... the proofs in IST may be regarded as abbreviations of proofs within conventional mathematics...''. In this purely syntactical approach, the author follows one of A. Robinson's original approaches which is characteristic by emphasizing new deductive procedures rather than new entities. The core of the paper is a reduction procedure for formulas and proofs of a theory \({\mathcal A}\) * \((=IST\) on a language enriched by the language of a model of a first order theory \({\mathcal A})\) into formulas and proofs of a theory \(\tilde {\mathcal A}\) \((=adequately\) enriched ZFC). This reduction procedure gives a solution to a problem posed by \textit{A. Robinson} [J. Symb. Logic 38, 500-516 (1973; Zbl 0289.02002), problem No.11] which consisted of finding ``... a purely syntactical transformation which correlates standard and nonstandard proofs of the same theorems in a large area...''.
reviewed by: Rade Živaljević
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 03C62
Mathematics Subject Classification ID: 03E70
zbMATH DE Number: 4045741
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: Internal Set Theory
MaRDI profile type: Publication
full work available at URL: 0168-0072(88)90050-4
OpenAlex ID: W2052045432
cites work: A self-avoiding random walk
cites work: Q5586197
cites work: Internal set theory: A new approach to nonstandard analysis
cites work: Non-standard analysis
cites work: Metamathematical problems
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description: scientific article; zbMATH DE number 4045741