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Some examples of use of transfinite induction in analysis

Nicola gigli

TL;DR

The paper introduces a transfinite, ordinal-indexed framework for proving existence of extremal objects in analysis by gradually enlarging a real-valued quantity at each ordinal stage. A key lemma shows that any monotone $\\\omega_1$-indexed process must stabilize within countably many steps, enabling existence proofs even when a direct real-valued objective is not readily quantified. It demonstrates the method through three applications: Hahn–Jordan decomposition, Ekeland's variational principle, and maximal globally hyperbolic development, including both a transfinite (small-steps) and an alternative (big-steps) approach to the MGHD problem. The work highlights DC$_{ω1}$ sufficiency and offers a viable substitute for Zorn-type arguments in settings where an explicit quantitative size can be defined.

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 $ω_1$ 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].

Some examples of use of transfinite induction in analysis

TL;DR

The paper introduces a transfinite, ordinal-indexed framework for proving existence of extremal objects in analysis by gradually enlarging a real-valued quantity at each ordinal stage. A key lemma shows that any monotone -indexed process must stabilize within countably many steps, enabling existence proofs even when a direct real-valued objective is not readily quantified. It demonstrates the method through three applications: Hahn–Jordan decomposition, Ekeland's variational principle, and maximal globally hyperbolic development, including both a transfinite (small-steps) and an alternative (big-steps) approach to the MGHD problem. The work highlights DC sufficiency and offers a viable substitute for Zorn-type arguments in settings where an explicit quantitative size can be defined.

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 to , 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].

Paper Structure

This paper contains 6 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Abstract presentation of the mechanism
  3. Three examples
  4. The Hahn-Jordan decomposition
  5. Ekeland's variational principle
  6. Maximal Globally Hyperbolic Development

Key Result

Lemma 2.1

Let $A$ be a set, $F\subset A$ a subset and $\mathcal{S}$ a non-empty collection of sequences of length $<\omega_1$ with the following properties: Then there is $(a_{\alpha'})_{\alpha'<\alpha}\in\mathcal{S}$ and $\bar{\alpha}<\alpha$ so that $a_{\bar{\alpha}}\in F$. In particular, $F$ is not empty.

Theorems & Definitions (10)

  • Lemma 2.1: Transfinite recursion over countable ordinals
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • Remark 3.7: The longest line