Inconsistent models for relevant arithmetics

Extract

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano's axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R^#, was absolutely consistent. It was pointed out that such a result escapes incautious formulations of Gödel's second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not ordinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.

The resulting model-theoretic structure (called in this paper RM3²) is interesting in its own right in that the set of sentences true therein constitutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R^#. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P^#. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3ⁱ here).