A majorizing semantics for hyperarithmetic sentences

A hyperarithmetic language $\mathbf L_\Lambda$ is considered, obtained by adding to the arithmetic language a special ternary predicate $H_\Lambda$ which acts as the “universal predicate” for $\mathbf L_\Lambda$ (for some scale of constructive ordinals $\Lambda$). The language $\mathbf L_\Lambda$ expresses a hierarchy $\{\Gamma_\alpha\}_{\alpha<\Lambda}$ of classes of formulas which is the constructive analog of the initial $\Lambda$-section of the classical hyperarithmetic hierarchy. Some properties of this hierarchy are introduced which give a convenient constructive theory $T_\Lambda$. 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 $\mathbf L_\Lambda$ 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 $\omega$–rule.