Perspex Machine VIII: axioms of transreal arithmetic

James A. D. W. Anderson; Norbert Völker; Andrew A. Adams

doi:10.1117/12.698153

29 January 2007 Perspex Machine VIII: axioms of transreal arithmetic

James A. D. W. Anderson, Norbert Völker, Andrew A. Adams

Proceedings Volume 6499, Vision Geometry XV; 649902 (2007) https://doi.org/10.1117/12.698153
Event: Electronic Imaging 2007, 2007, San Jose, CA, United States

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.

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James A. D. W. Anderson, Norbert Völker, and Andrew A. Adams "Perspex Machine VIII: axioms of transreal arithmetic", Proc. SPIE 6499, Vision Geometry XV, 649902 (29 January 2007); https://doi.org/10.1117/12.698153

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