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Proceedings Paper

Perspex Machine VIII: axioms of transreal arithmetic
Author(s): James A. D. W. Anderson; Norbert Völker; Andrew A. Adams
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Paper Abstract

Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. It allows the specification of the Universal Perspex Machine which unifies geometry with the Turing Machine. Here we axiomatise the algebraic structure of transreal arithmetic so that it provides a total arithmetic on any appropriate set of numbers. This opens up the possibility of specifying a version of floating-point arithmetic that does not have any arithmetical exceptions and in which every number is a first-class citizen. We find that literal numbers in the axioms are distinct. In other words, the axiomatisation does not require special axioms to force non-triviality. It follows that transreal arithmetic must be defined on a set of numbers that contains{-∞,-1,0,1,∞,&pphi;} as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

Paper Details

Date Published: 29 January 2007
PDF: 12 pages
Proc. SPIE 6499, Vision Geometry XV, 649902 (29 January 2007); doi: 10.1117/12.698153
Show Author Affiliations
James A. D. W. Anderson, Univ. of Reading (United Kingdom)
Norbert Völker, Univ. of Essex (United Kingdom)
Andrew A. Adams, Univ. of Reading (United Kingdom)

Published in SPIE Proceedings Vol. 6499:
Vision Geometry XV
Longin Jan Latecki; David M. Mount; Angela Y. Wu, Editor(s)

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