Perspex Machine VIII: axioms of transreal arithmetic
| Autor: | Andrew A. Adams, Norbert Völker, James Anderson |
|---|---|
| Rok vydání: | 2007 |
| Předmět: |
Discrete mathematics
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES True arithmetic Second-order arithmetic Elementary arithmetic Arbitrary-precision arithmetic Robinson arithmetic Arithmetic function Primitive recursive arithmetic Hardware_ARITHMETICANDLOGICSTRUCTURES Arithmetic Non-standard model of arithmetic Mathematics |
| Zdroj: | SPIE Proceedings. |
| ISSN: | 0277-786X |
| DOI: | 10.1117/12.698153 |
| Popis: | 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,∞,p} as a proper subset. We note that the axioms have been shown to be consistent by machine proof. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|