Arithmetic Formulated Relevantly


 The purpose of this paper is to formulate first-order Peano arithmetic within the resources relevant logic, and demonstrate certain properties system thus formulated. Striking among these are facts that (1) it trivial absolutely consistent, but (2) classical straightforwardly contained in arithmetic. Under (1), I shall show particular 0 = 1 a non-theorem arithmetic; this, course, exactly formula whose unprovability was sought Hilbert program for proving consistent. (2), exhibit requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, critical problem whether Ackermann’s rule γ admissible theories featured be called R♯. Its logical base R implication, taken its form RQ. Among other arithmetics we consider here systems C♯, J♯, RM3♯; based respectively on logic C, intuitionistic J, Sobocinski-Dunn semi-relevant RM3. And another feature will presentation natural deduction R♯, along lines valid general. This formulation R♯ makes possible construct relevantly arithmetical deductions an easy way; on, respects more convenient than, formulations logics developed by Anderson Belnap Entailment.