1. Arai, T.: Provably $$\Delta ^{0}_{2}$$ and weakly descending chains. In: Arai, T. Feng, Q., Kim, B., Wu, G., Yang, Y. (eds.) Proceedings of the 11th Asian Logic Conference, pp. 1–21. World Scientific (2011)
2. Arai, T., Mints, G.: Extended normal form theorems for logical proofs from axioms. Theor. Comput. Sci. 232, 121–132 (2000)
3. Barwise, J.: Admissible Sets and Structures. Springer, Berlin (1975)
4. Beth, E.W.: On Padoa’s method in the theory of definition. Indag. Math. 15, 330–339 (1953)
5. Beth, E.W.: Semantic entailment and formal derivability. Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen (Amsterdam), Afdeling Letterkunde, Nieuwe Reeks 18, pp. 309–342 (1955)