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

We consider the Double-negation Shift (DNS) as a constructive principle in its own right and its effect on modified realizability (MR) and Dialectica (D) interpretations. We notice that DNS proves its own MR-interpretation, meaning that a priori one does not have to consider the more complex D-interpretation with Bar Recursion for interpreting Analysis. From the “with truth” variant of MR, we obtain the closure under Weak Church’s Rule. We notice, in contrast, that DNS proves the double negation of the Limited Principle of Omniscience (LPO), hence of the solvability of the Halting Problem, and recall the related fact that DNS refutes a formal version of Church’s Thesis (CT). This shows that intuitionistic Arithmetic plus DNS presents a distinct variant of Constructive Mathematics. We revisit the standard approach of Kreisel and Spector for showing consistency of Analysis, and show that one can omit the preliminary double-negation translation phase. Finally, we formalize the proofs of not-not-LPO and notCT using a previously introduced constructive logic based on delimited control operators from the theory of programming languages.