Diagonal arguments and fixed points

Karimi, A.; Salehi, S.

Diagonal arguments and fixed points

Document Type : Research Paper

Authors

A. Karimi ¹
S. Salehi ²

¹ Department of Mathematical Sciences‎, ‎Tarbiat Modares University‎, ‎P.O‎. ‎Box 14115--134‎, ‎Tehran‎, ‎Iran.

² Research Institute for Fundamental Sciences‎, ‎University of Tabriz‎, ‎P.O‎. ‎Box 51666--16471‎, ‎Tabriz‎, ‎Iran‎.

Abstract

‎A universal schema for diagonalization was popularized by N.S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fit more theorems in the universal‎
‎schema of diagonalization‎, ‎such as Euclid's proof for the infinitude of the primes and new proofs of G.~Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset‎. ‎Then‎, ‎in Linear Temporal Logic‎, ‎we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox (1985‎, ‎1993)‎. ‎Thus‎, ‎Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic‎. ‎Again the diagonal schema of the paper is used in this proof; and it is also shown that G.~Priest's inclosure schema (1997) can fit in our universal diagonal/fixed-point schema‎. ‎We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions‎, ‎such as primitive recursive functions) in the schema.

Keywords

Main Subjects

03-XX Mathematical logic and foundations

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