Some examples of use of transfinite induction in analysis

TL;DR

This paper explores a novel approach using transfinite induction over countable ordinals to establish the existence of extremal objects in analysis, including applications in General Relativity.

Contribution

It introduces a method that employs ordinal-indexed recursive procedures to prove existence results without relying solely on real-valued maximization functions.

Findings

The method can prove the existence of maximal globally hyperbolic developments.

It offers an alternative to traditional Dezornification techniques.

The approach is applicable even when measuring maximality via real-valued functions is not straightforward.

Abstract

It is not uncommon in analysis that existence of extremal objects is obtained via an iterative procedure: we start from a given admissible object, then modify it, then modify again etc... If being extremal means maximimizing a real valued quantity and we are sure to approach the supremum fast enough, after a countable number of steps and a limiting procedure we are done. In this short note we want to advertise a slightly different line of thought, where rather than trying to approach the supremum fast enough, we: try to increase, if possible, the function to be maximized and, at the same time, index our recursive procedure over ordinals. Since there are no increasing functions from $\omega_1$ to $\R$, the procedure must stop at some countable ordinal and existence is proved anyway. The advantage of this line of reasoning is that it can be helpful even in situations where it is not so evident how to measure `being maximal' via a real valued function. This is the case, for instance, for existence of a Maximal Globally Hyperbolic Development of an initial data set in General Relativity. Speaking of this particular example, we also show that such `real-valued quantification' of the size of a development is actually possible, thus existence of a maximal one can be obtained in a countable number of steps using the original argument in [2] together with the standard procedure depicted above. This provides a way alternative to the one given in [5] to `dezornify' the proof in [2].