 Relevant logic has been used as the basis for mathematical theories other than set theory. Meyer has produced a variation of Peano arithmetic based on the logic \(\mathbf{R}\). Meyer gave a finitary proof that his relevant arithmetic does not have \(0 = 1\) as a theorem. Thus Meyer solved one of Hilbert’s central problems in the context of relevant arithmetic; he showed using finitary means that relevant arithmetic is absolutely consistent. This makes relevant Peano arithmetic an extremely interesting theory. Unfortunately, as Meyer and Friedman have shown, relevant arithmetic does not contain all of the theorems of classical Peano arithmetic. Hence we cannot infer from this that classical Peano arithmetic is absolutely consistent (see Meyer and Friedman 1992).