A note on non-classical nonstandard arithmetic

Abstract

Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic, have been proposed. We single out one particular system by Dinis–Gaspar, which is categorised by the authors as being part of intuitionistic nonstandard arithmetic. Their system is indeed inconsistent with the Transfer axiom of Nonstandard Analysis, and the latter axiom is classical in nature as it implies (higher-order) comprehension. Inspired by this observation, the main aim of this paper is to provide answers to the following questions:(Q1)In the spirit of Reverse Mathematics, what is the minimal fragment of Transfer that is inconsistent with the Dinis–Gaspar system?(Q2)What other axioms are inconsistent with the Dinis–Gaspar system? Our answer to the first question suggests that the aforementioned inconsistency actually derives from the axiom of extensionality relative to the standard world, and that other (much stronger) consequences of Transfer are actually harmless. Perhaps surprisingly, our answer to the second question shows that the Dinis–Gaspar system is inconsistent with a number of (non-classical) continuity theorems which one would – in our opinion – categorise as intuitionistic in the sense of Brouwer. Finally, we show that the Dinis–Gaspar system involves a standard part map, suggesting this system also pushes the boundary of what still counts as ‘Nonstandard Analysis’ or ‘internal set theory’.