# Diagonal arguments and fixed points

*Bulletin of the Iranian Mathematical Society*43 (5):1073-1088 (2017)

**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.

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