Abstract
In this thesis, we adapt several prominent methods to state consistent axiomatic theories of (type-free) arithmetical truth for the particular levels and the entire Grzegorczyk (primitive recursive) hierarchy and arithmetical hierarchy. More specifically, the considered theories include: (1) a theory of naive truth for some basic level of the Grzegorczyk hierarchy; (2) theories of Friedman-Sheard truth for higher levels of the Grzegorczyk hierarchy; (3) theories of Kripke-Feferman truth, grounded truth, and disquotational truth for the entire Grzegorczyk hierarchy; (4) iterations and progressions (via so-called reflection schemas) of theories of Kripke-Feferman truth for the particular levels and the entire arithmetical hierarchy. The bulk of our work consists in proving -- upon devising the required technical machinery and concepts -- that the considered theories of truth are (in the specific proof-theoretic sense to be devised) interpretable in corresponding theories of arithmetic by employing recursion-theoretic tools. A great deal of our proof-theoretic analysis relies on Parsons' theorem.
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