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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>, 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 [<CitationRef CitationID="CR3">3</CitationRef>] together with the standard procedure depicted above. This provides a way alternative to the one given in [<CitationRef CitationID="CR9">9</CitationRef>] to ‘dezornify’ the proof in [<CitationRef CitationID="CR3">3</CitationRef>].</p>

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Some examples of use of transfinite induction in analysis

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Nicola Gigli

摘要

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 \(\omega _1\) ω 1 to \(\mathbb {R}\) 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 [3] together with the standard procedure depicted above. This provides a way alternative to the one given in [9] to ‘dezornify’ the proof in [3].