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Computability of sets in Euclidean space

Computability of sets in Euclidean space AbstractWe consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝn and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Pure Mathematics and Applications de Gruyter

Computability of sets in Euclidean space

Potgieter, Petrus H
Pure Mathematics and Applications , Volume 30 (3): 13 – Oct 1, 2022
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Publisher
de Gruyter
Copyright
© 2022 Petrus H Potgieter, published by Sciendo
eISSN
1788-800X
DOI
10.2478/puma-2022-0025
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Abstract

AbstractWe consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝn and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.

Journal

Pure Mathematics and Applications – de Gruyter

Published: Oct 1, 2022

Keywords: Computability of sets; separability; computable function; impossibility results; Mandelbrot set; 03D80; 03D78

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