and survey -- A formal system of classical analysis -- Elimination of extensionality -- Translation of classical into intuitionistic approximated theories -- Gödel's functional interpretation in the narrower sense -- The calculus T of the primitive recursive functionals -- Functional interpretation of classical arithmetic plus (ER)-qf, (AC)-qf and functional interpretation in the narrower sense of Heyting-analysis plus (ER)-qf, (MP), in T -- The calculus T?BR of the bar recursive functionals -- Functional interpretation of classical (AC)o-, (?AC)-analysis with (ER)-qf and functional interpretation in the narrower sense of Heyting-analysis plus (ER)-qf, (MP), in T?BR -- Further consequences from the functional interpretation of classical analysis -- Consistency proof by computation. Computation of T?BRo...o?? -- Generalized inductive definitions -- Generalization of bar induction BID and the inductive generation processes to trees over species -- A model for T?BR -- On the bar recursive model of classical analysis and the general bar induction over species.