“School of Mathematics”
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Paper IPM / M / 17377  


Abstract:  
Interesting as they are by themselves in philosophy and mathematics, paradoxes can be made even more fascinating when turned into proofs and theorems. For example, RussellÃ¢??s paradox, which overthrew FregeÃ¢??s logical edifice, is now a classical theorem in set theory, to the effect that no set contains all sets. Paradoxes can be used in proofs of some other theoremsÃ¢??thus LiarÃ¢??s paradox has been used in the classical proof of TarskiÃ¢??s theorem on the undefinability of truth in sufficiently rich languages. This paradox (as well as RichardÃ¢??s paradox) appears implicitly in GÃÂ¶delÃ¢??s proof of his celebrated first incompleteness theorem. In this paper, we study YabloÃ¢??s paradox from the viewpoint of first and secondorder logics. We prove that a formalization of YabloÃ¢??s paradox (which is second order in nature) is nonfirstorderizable in the sense of George Boolos (1984).
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