Interpretability in Robinson's Q (Q2870105)

\textit{E. Nelson} defends in his book [Predicative arithmetic. Princeton, New Jersey: Princeton University Press (1986; Zbl 0617.03002)] a strong formalistic position in mathematics that doubts, or even denies, the totality of exponentiation. This leads him to take refuge in Raphael Robinson's arithmetic Q. That drove Nelson to investigate the amount of arithmetic that can be interpreted in Q, a project in which he was followed by others, resulting in a variety of results about interpretability (or non-interpretability) in Q.NEWLINENEWLINEFernando Ferreira and Gilda Ferreira's article collects and discusses some nice results about interpretability (or non-interpretability) in Q. For example, two results that, in a way, substantiate Nelson's position: one from Alex Wilkie saying that the totality of exponentiation cannot be interpreted in Q, and another one from Robert Solovay saying that the negation of the totality of exponentiation can be interpreted in Q. Another example, two results that, in a way, undermine Nelson's position: one from Robert Solovay saying that there are two formulas such that they are both interpretable in Q but their conjunction is not, and another one (more general) also from Robert Solovay saying that there is a formula such that it and its negation are both interpretable in Q. Another example, an original result of Fernando Ferreira and Gilda Ferreira, is that the theory of analysis BTPSA (associated to the polyspace computable functions) can be interpreted in Q, which is somewhat unexpected because BTPSA is sufficiently strong to establish Riemann integration all the way through to the fundamental theorem of calculus.