The Berry Paradox

Abstract Berry’s Paradox, like Russell’s Paradox, is a ‘paradox’ in name only. It differs from genuine logico-semantic paradoxes such as the Liar Paradox, Grelling’s Paradox, the Postcard Paradox, Yablo’s Paradox, the Knower Paradox, Prior’s Intensional Paradoxes, and their ilk. These latter arise from semantic closure. Their genuine paradoxicality manifests itself as the non-normalizability of the formal proofs or disproofs associated with them. The Russell, the Berry, and the Burali-Forti ‘paradoxes’, by contrast, simply reveal the straightforward inconsistency of their respective existential claims—that the Russell set exists; that the Berry number exists; and that the ordinal of the well-ordering of all ordinals exists. The disproofs of these existential claims are in free logic and are in normal form. They show that certain complex singular terms do not—indeed, cannot —denote. All this counsels reconsideration of Ramsey’s famous division of paradoxes and contradictions into his Group A and Group B. The proof-theoretic criterion of genuine paradoxicality formally explicates an informal and occasionally confused notion. The criterion should be allowed to reform our intuitions about what makes for genuine paradoxicality, as opposed to straightforward (albeit surprising) inconsistency.