On a Spector ultrapower of the Solovay model

We prove that a Spector--like ultrapower extension \gN of a countable Solovay model \gM (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension \gM[\al] where \al is a random real over \gM. The proof involves an almost everywhere uniformization theorem in the Solovay model

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis