Perspex Machine VIII: axioms of transreal arithmetic

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

Perspex Machine IX: transreal analysis

“…This is summarised in the following table. The table is consistent with the products obtained via the axioms of transreal arithmetic 4 or, for that matter, via real arithmetic. The table contains real numbers and .…”

“…All such counter proofs are falsified or rendered irrelevant by the machine proof of consistency of transreal arithmetic. 4 Nonetheless, it is useful to examine these counter proofs to understand the objections to division by zero that have been held for so very long.…”

“…And why should we use these incoherent extensions rather than coherent ones? Carrying out the evaluation in transreal arithmetic 4 produces the unproblematic tautology: .…”

“…It remains to be shown that the addition theorem holds when any or all of are strictly transreal. As transreal addition and multiplication are commutative 4 we need consider only strictly transreal and arbitrarily transreal. As usual, , denotes the set of transreal numbers.…”

“…The new arithmetic is axiomatised and shown to be consistent in. 4 Transreal arithmetic is a total arithmetic, which means that every arithmetical operation can be applied to any number such that the result is a number. This has consequences for real analysis.…”