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The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overviewof recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

for insightful and helpful comments. I am indebted to Ali Enayat for invaluable communications and information about the theory of satisfaction classes, and to Neil Barton and Sy-David Friedman for answering my questions about their article. I am deeply grateful to the two anonymous referees for very extensive comments and helpful suggestions, which led to a considerable improvement of the article. Finally, I would like to express my special thanks to Leon Horsten and Philip Welch for immensely fruitful discussions on the topic of this article. 2 We use the term 'predicative' in the (literal) sense of "having the quality of predicating something" or "forming or having the function of a predicate" (OED) here and thus call the view at issue "predicativism." The term is, however, often used in philosophy and logic with a technical (figurative) meaning, according to which something is called predicative if it is defined, expressed, or constructed without quantification over domains that include it. If one wants to reserve the term 'predicative' for the latter technical use, one could alternatively call the view "predicationalism" or "predicational interpretation of classes." 3 Charles Parsons, "Sets and Classes," in Mathematics in Philosophy (Ithaca: Cornell University Press, 1983), pp. 209-220.

This paper studies several systems of the transfinite iteration and autonomous progression of self-applicable truth and determines their proof-theoretic strength.