TAPP is a total applicative theory, conservative over intuitionistic arithmetic. In this paper, we first show that the same holds for TAPP+ the choice principle EAC; then we extend TAPP with choice sequences and study the principle EBIa0 . The resulting theories are used to characterise the arithmetical fragment of EL +EBIa0. As a digression, we use TAPP to show that P. Martin-Löf's basic extensional theory ML0 is conservative over intuitionistic arithmetic.

Goodman's theorem states that intuitionistic arithmetic in all finite types plus full choice, HA + AC, is conservative over first-order intuitionistic arithmetic HA. We show that this result does not extend to various subsystems of HA, HA with restricted induction.

There are two different ways to introduce the notion of truthin constructive mathematics. The first one is to use a Tarskian definition of truth in aconstructive (meta)language. According to some authors, (Kreisel, van Dalen, Troelstra ... ),this definition is entirely similar to the Tarskian definition of classical truth (thesis A).The second one, due essentially to Heyting and Kolmogorov, and known as theBrouwer–Heyting–Kolmogorov interpretation, is to explain informally what it means fora mathematical proposition to be constructively proved. According to other authors (...) (Martin-Löfand Shapiro), this interpretation and the Tarskian definition of truth amount to thesame (thesis B). My aim in this paper is to show that thesis A is only reasonable, that thesis Bis false and to answer the following question: what is defined by the Tarskian definition ofconstructive truth? (shrink)