Consistency of the minimalist foundation with Church thesis and Bar Induction

We consider a version of the minimalist foundation previously introduced to formalize predicative constructive mathematics. This foundation is equipped with two levels to meet the usual informal practice of developing mathematics in an extensional set theory (its extensional level) with the possibility of formalizing it in an intensional theory enjoying a proofs as programs semantics (its intensional level). For the intensional level we show a realizability interpretation validating Bar Induction and formal Church thesis for type-theoretic functions. This is possible because in our foundation the well-known result by Kleene that Brouwer’s principle of Bar Induction is inconsistent with the formal Church thesis for choice sequences can be decomposed as follows: Brouwer’s Bar Induction, where choice sequences are functional relations, is inconsistent with the formal Church thesis for type-theoretic functions (from natural numbers to natural numbers) and the axiom of unique choice transforming a functional relation between natural numbers into a type-theoretic function. As a consequence this model disproves the validity of the axiom of unique choice in our foundation. This model can serve to interpret the whole foundation in a classical predicative set theory by keeping the computational interpretation of predicative sets as data types and their type-theoretic functions as programs. Moreover it shows that choice sequences of Cantor space, those of Baire space, and real numbers both as Dedekind cuts or Cauchy sequences, do not form a set in the minimalist foundation. MSC 2000: 03G30 03B15 18C50 03B20 03F55