On the mathematical and foundational significance of the uncountable

Por • 5 jul, 2020 • Sección: Filosofía

Dag Normann, Sam Sanders

For various historical, practical, and foundational reasons, large parts of mathematics are studied indirectly via countable approximations, also called codes. It is a natural question whether this indirect study of codes is faithful to the original development in mathematics, or whether approximations somehow distort the latter. Another natural question is which parts of basic mathematics can(not) be studied via these representations. In this paper, we formulate new answers to these old questions. Our answers stem both from mathematics itself (via the study of the gauge integral) and its foundations (via Hilbert-Bernays’ Grundlagen der Mathematik and its spiritual successor Reverse Mathematics). We identify a number of basic theorems from (uncountable) mathematics for which the logical and computational properties are completely (and even maximally) distorted upon introducing countable approximations. In a nutshell, while countable approximations are interesting and important, even extremely basic ‘uncountable’ mathematics is infinitely more complicated than the ‘countable picture’ involving codes suggests.

 arXiv:1711.08939v1 [math.LO]

 Logic (math.LO)

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