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Supertasks, physics and the Axiom of Infinity

Author:

Leon, Antonio

Category:

Essays

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

October 03, 2021

File:

/Essays-Mathematics and Applied Mathematics/Download/8206

Downloads:

277

Keywords:

actual infinity, potential infinity, supertask

Abstract:

It seems reasonable to assume that mathematical infinity was not the objective of Zeno’s Dichotomy (in any of its variants), however, a sort of mathematical infinity was already present in these celebrated arguments. Aristotle proposed a first solution to Zeno’s Dichotomy by introducing what we now call one-to-one correspondences, the key instrument of modern infinitist mathematics. But Aristotle, more naturalist than platonic, finally rejected the method of pairing the elements of two infinite collections (in this case of points and instants) and introduced instead the distinction between actual and potential infinities. Aristotle’s distinction served to define, gross modo, two opposite positions on the nature of infinity for more than twenty centuries. The actual infinity was finally mathematized through set theory in the first years of the XX century and the discussions on its potential or actual nature almost vanished. But, as we will see here, things still remain to be said on this issue.