Notre Dame Journal of Formal Logic 35 (2):204-218 (1994)
|Abstract||In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of deﬁnition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a diﬀerent perspective on mathematically more “respectable” entities. Revision Rules were ﬁrst introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they have been further developed to provide the foundations for a general theory of (possibly circular) deﬁnitions. Revision Rules are non-monotonic inductive operators that are iterated into the transﬁnite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then deﬁne a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta|
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