Carnap's remarks on Impredicative Definitions and the Genericity Theorem (1997)
| Venue: | IN LOGIC, METHODOLOGY AND PHILOSOPHY OF SCIENCE: LOGIC IN FLORENCE |
| Citations: | 1 - 1 self |
BibTeX
@INPROCEEDINGS{Fruchart97carnap'sremarks,
author = {Thomas Fruchart and Giuseppe Longo},
title = {Carnap's remarks on Impredicative Definitions and the Genericity Theorem },
booktitle = {IN LOGIC, METHODOLOGY AND PHILOSOPHY OF SCIENCE: LOGIC IN FLORENCE},
year = {1997},
publisher = {Kluwer}
}
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Abstract
In a short, but relevant paper [Car31], Rudolf Carnap summarizes the logicist foundation of mathematics, largely following Frege and Russell 's view. Carnap moves away though from Russell's approach on a crucial aspect: a detailed justification of impredicative definitions (a formal version of Russell's "vicious circle"), that he accepts. In this note we revisit Carnap's justification of impredicativity, within the frame of impredicative Type Theory. More precisely, we recall the treatment of impredicativity given in Girard's System F and justify it by reference to a recent result, the Genericity Theorem in [LMS93], which may help to set on mathematical grounds Carnap's informal remark. We then discuss the logical complexity of (the proof of) that theorem. Finally, the role of the Genericity Theorem in understanding the surprising "uniformities" of the consistency proof of Arithmetic, via System F, is hinted. The problem A definition is said to be impredicative, if it defines a concep...
