Robinson arithmetic, denoted as \( R \), is a weak system of arithmetic that is part of the field of mathematical logic and foundational studies. It was introduced by the mathematician and logician John Robinson in the 1950s. The key features of Robinson arithmetic include: 1. **Language**: The language of Robinson arithmetic includes a number of basic symbols for logical operations (like conjunction and disjunction), equality, and a unary function symbol typically interpreted as a successor function.
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Robinson arithmetic is a finitely axiomatized first-order theory in the language . Its seven axioms are the universal closures ofThey describe successor, addition and multiplication without an induction schema. Each calculation on particular standard numerals can still be verified: repeated use of the recursion equations proves, for example, . This finite computational strength is enough for representing computable finite calculations and for the diagonal lemma used in crude incompleteness theorem. Peano arithmetic adds induction for arbitrary arithmetic formulas.[ref]
Introduction to the Theory of Computation: Computability, Complexity, and the Lambda Calculus by Jean Gallier and Jocelyn Quaintance (2020). Chapter6, printed p.156, seven axioms and induction schema