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Transfinite recursive progressions of axiomatic theories

Published online by Cambridge University Press:  12 March 2014

Solomon Feferman
Solomon Feferman*
Affiliation:
Stanford University, Institute for Advanced Study
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Extract

The theories considered here are based on the classical functional calculus (possibly of higher order) together with a set A of non-logical axioms; they are also assumed to contain classical first-order number theory. In foundational investigations it is customary to further restrict attention to the case that A is recursive, or at least recursively enumerable (an equivalent restriction, by [1]). For such axiomatic theories we have the well-known incompleteness phenomena discovered by Godei [6]. Quite far removed from such theories are those based on non-constructive sets of axioms, for example the set of all true sentences of first-order number theory. According to Tarski's theorem, there is not even an arithmetically definable set of axioms A which will give the same result (cf. [18] for exposition).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 3 , September 1962 , pp. 259 - 316
Copyright
Copyright © Association for Symbolic Logic 1962

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