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Paper   IPM / M / 17377
School of Mathematics
  Title:   `Sometime a Paradox', now proof: Yablo is not first order
  Author(s):  Saeed Salehi
  Status:   Published
  Journal: Log. J. IGPL
  Vol.:  30
  Year:  2022
  Pages:   71-77
  Supported by:  IPM
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, Russellâ??s paradox, which overthrew Fregeâ??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 theoremsâ??thus Liarâ??s paradox has been used in the classical proof of Tarskiâ??s theorem on the undefinability of truth in sufficiently rich languages. This paradox (as well as Richardâ??s paradox) appears implicitly in Gödelâ??s proof of his celebrated first incompleteness theorem. In this paper, we study Yabloâ??s paradox from the viewpoint of first- and second-order logics. We prove that a formalization of Yabloâ??s paradox (which is second order in nature) is non-first-orderizable in the sense of George Boolos (1984).

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