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 as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

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In real-world applications, Internet

Abstract. This paper introduces the logical system HOL2P that extends classical higher order logic (HOL) with type operator variables and universal types. HOL2P has explicit term operations for type abstraction and type application. The formation of type application terms t [ T ] is restricted to small types T that do not contain any universal types. This constraint ensures the existence of a set-theoretic model and thus consistency.The expressiveness of HOL2P allows category-theoretic concepts such as natural transformations and initial algebras to be applied at the level of polymorphic HOL functions. The parameterisation of terms with type operators adds genericity to theorems. Type variable quantification can also be expressed.A prototype of HOL2P has been implemented on top of HOL-Light. Type inference is semi-automatic, and some type annotations are necessary. Reasoning is supported by appropriate tactics. The implementation has been used to check some sample derivations.

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