 In recent years, logicians have investigated many other systems of arithmetic that are weaker than Peano Arithmetic. Often these logicians took their inspiration from computability issues, for example the study of functions computable in polynomial time. They have given a partial answer to the question: “For which theories of arithmetic does Solovay’s arithmetical completeness theorem (with respect to the appropriate provability predicate) still hold?” To discuss this question, two concepts are needed. \(\Delta_0\)-formulas are arithmetical formulas in which all quantifiers are bounded by a term, for example