A majorizing semantics for hyperarithmetic sentences

A hyperarithmetic language is considered, obtained by adding to the arithmetic language a special ternary predicateH which acts as the universal predicate for (for some scale of constructive ordinals ). The language expresses a hierarchy {}_< of classes of formulas which is the constructive analog of the initial -section of the classical hyperarithmetic hierarchy. Some properties of this hierarchy are introduced which give a convenient constructive theoryT. It is shown that the majorizing semantics introduced in [1] (for an equivalent variant see [2]) can be extended to the sentences of the language for sentences of the arithmetic language. The basis for the construction of the majorant is the idea (stated in [2]) of relating the majorant to deducibility in systems with an -rule.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 30–37, 1977.