 Robert K. Meyer (1976) seems to have been the first to think of an inconsistent arithmetical theory. At this point, he was more interested in the fate of a consistent theory, his relevant arithmetic R#. This amounts to the axioms for Peano arithmetic, with a base of the quantified relevant logic RQ, and Meyer hoped that the weaker base of relevant logic would allow more models. He was right. There proved to be a whole class of inconsistent arithmetical theories; see Meyer and Mortensen (1984), for example. In a parallel with the above remarks on rehabilitating logicism, Meyer argued that these arithmetical theories provide the basis for a revived Hilbert Program. Hilbert’s program was the project of rigorously formalising mathematics and proving its consistency by simple finitary/inductive procedures. It was widely held to have been seriously damaged by Gödel’s Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer’s construction was that within his arithmetic R# it was demonstrable by finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert’s goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable as long as inconsistency-tolerant logics are used.