We present a proof of Goodmanʼs Theorem, which is a variation of the proof of Renaldel de Lavalette [9]. This proof uses in an essential way possibly divergent computations for proving a result which mentions systems involving only terminating computations. Our proof is carried out in a constructive metalanguage. This involves implicitly a covering relation over arbitrary posets in formal topology, which occurs in forcing in set theory in a classical framework, but can also be defined constructively.

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.