A Direct Version of Veldman\u27s Proof of Open Induction on Cantor Space via Delimited Control Operators

Abstract

First, we reconstruct Wim Veldman\u27s result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov\u27s Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov\u27s Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski