Diagonalizing by Fixed-Points

TL;DR

This paper extends a universal diagonalization schema to include classical theorems, paradoxes, and fixed-point results across logic and mathematics, demonstrating its broad applicability and revealing new insights such as the logical status of Yablo's paradox.

Contribution

It generalizes the diagonalization schema to encompass more theorems and paradoxes, including Euclid's theorem, Cantor's theorem, and Yablo's paradox, showing its unifying power.

Findings

Diagonality schema applies to Euclid's theorem and Cantor's theorem.

Yablo's paradox is shown as a theorem in Linear Temporal Logic.

Existence of Ackermann-like dominating functions is established.

Abstract

A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) 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 theorem on the infinitude of the primes and new proofs of 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. 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 also it is 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---like primitive recursives) using the schema.