Diagonalizing by Fixed-Points

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.

💡 Research Summary

The paper revisits the universal diagonal‑fixed‑point schema originally popularized by Noson Yanofsky (2003). The core observation is that if a function α on a set D has no fixed point, then for any set B and any binary function f : B × B → D the derived function g(x) = α(f(x,x)) cannot coincide with any “section” f(·,b) of f. In other words, the existence of a diagonalized‑out object forces the existence of a fixed point for α; the contrapositive yields a powerful method for proving non‑surjectivity or non‑injectivity of various maps.

The authors systematically apply this schema to a collection of classical results that are not usually presented as diagonal arguments.

1. Euclid’s theorem on the infinitude of primes – They define a predicate f(n,m) that is true exactly when every prime divisor of n! + 1 is smaller than m. Using the negation function as α, the diagonal function g(n)=¬f(n,n) is constantly true. If a bound p existed for all primes, then f(p,p)=¬f(p,p) would give a fixed point of ¬, contradicting the hypothesis. Hence there can be no such bound, i.e., infinitely many primes.

2. Boolos’s proof of Cantor’s theorem – For a set A and a function h : P(A)→A they construct the set D_h = {a ∈ A | ∃Y⊆A