  In 2002 Robert Solovay proved that a subsystem BI of classical second order
arithmetic, with bar induction and arithmetical countable choice, can be
negatively interpreted in the neutral subsystem BSK of Kleene's intuitionistic
analysis FIM using Markov's Principle MP. Combining this result with Kleene's
formalized recursive realizability, he established (in primitive recursive
arithmetic PRA) that FIM + MP and BI have the same consistency strength.
  This historical note includes Solovay's original proof, with his permission,
and the additional observation that Markov's Principle can be weakened to a
double negation shift axiom consistent with Brouwer's creating subject
counterexamples.

Logic;History and Overview