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Metastability and Higher-Order Computability

  • Sam SandersEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10703)

Abstract

Tao’s notion of metastability is classically equivalent to the well-known ‘epsilon delta’ definition of Cauchy sequence, but the former has much nicer computational properties than the latter. In particular, results from the proof mining program suggest that metastability gives rise to highly uniform and computable results under very general conditions, in contrast to the non-uniform and/or non-computable rates of convergence emerging from basic convergence theorems involving the definition of Cauchy sequence. As a trade-off for these high levels of uniformity, metastability only provides information about a finite but arbitrarily large domain. In this paper, we apply this ‘metastability trade-off’ to basic theorems of mathematics, i.e. we introduce in the latter finite domains in exchange for high(er) levels of uniformity. In contrast to the aforementioned effective results from proof mining, we obtain functionals of extreme computational hardness. Thus, our results place a hard limit on the generality of the metastability trade-off and show that metastability does not provide a ‘king’s road’ to computable mathematics. Perhaps surprisingly, we shall make use of Nonstandard Analysis (NSA) to establish the aforementioned results (which do not involve NSA).

Keywords

Metastability Higher-order computability Nonstandard analysis 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Center for Advanced Studies and Munich Center for Mathematical PhilosophyLMU MunichMunichGermany

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