Yablifying the Rosser Sentence
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In a recent paper (Cieśliński and Urbaniak 2012), Urbaniak and Cieśliński describe an analogue of the Yablo Paradox, in the domain of formal provability. Just as the infinite sequence of Yablo sentences inherit the paradoxical behavior of the liar sentence, an infinite sequence of sentences can be constructed that inherit the distinctive behavior of the Gödel sentence. This phenomenon—the transfer of the properties of self-referential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—suggests a number of interesting logical questions. The purpose of this paper is to give a precise statement of a conjecture from Cieśliński and Urbaniak (2012) regarding the unwinding of the Rosser sentence, and to demonstrate that this precise statement is false. We begin with some preliminary motivation, introduce the conjecture against the background of some related results, and finally, in the last section, move on to the proof, which adapts a method used by Solovay and Guaspari.
KeywordsParadoxes Incompleteness Truth Yablo’s paradox Gödel’s theorem Rosser’s theorem Self-reference Well-foundedness
Thanks to Chris Porter and an anonymous reviewer for helpful comments and editorial suggestions; thanks to Tim Bays, Mic Detlefsen, Curtis Franks, Chris Porter, Tony Strimple and Sean Walsh for invaluable discussion and advice. This research was partially funded by EMSW21-RTG-0838506.
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