Adding proof objects and inductive definition mechanisms to frege structures
A constructive theory RPT (Reflective Proof Theory) of proofs which has the following three features is introduced. (1) Proofs as objects. (2) Hierarchies of propositions and truths. (3) The mechanisms of inductive definitions of predicates. Three kinds of structures called Frege structures with inductively defined predicates, Frege structures with proof objects and proof structures are also introduced. These structures are obtained by generalizing certain aspects of RPT and they are all closely related to Frege structures.
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- Aczel, P., Frege structures and the notions of proposition, truth and set, pp. 31–59 in The Kleene Symposium, Barwise, J., Keisler, H.J., Kunen, K. (eds.), North-Holland, 1980.Google Scholar
- Beeson, M.J., Foundations of Constructive Mathematics, Springer-Verlag, 1985.Google Scholar
- Feferman, S., A language and axioms for explicit mathematics, pp. 87–139 in Algebra and Logic, Lect. Notes in Math. 450, Crossley, J.N. (ed.), Springer-Verlag, 1975.Google Scholar
- Kobayashi, S., Consistency of Beeson's formal system RPS and some related results, pp. 120–140 in Mathematical Logic and Applications, Lect. Notes in Math. 1388, Shinoda, J., Slaman, T.A. and Tugué, T. (eds.), Springer-Verlag, 1987.Google Scholar
- Martin-Löf, P., Intuitionistic Type Theory, Bibliopolis, 1984.Google Scholar
- Sato, M. and Kameyama, Y., Constructive Programming in SST, pp. 23–30 in Proceedings of the Japanese-Czechoslovak Seminar on Theoretical Foundations of Knowledge Information Processing (ed. Arikawa, S. and Vlach, M.), Inorga, 1990.Google Scholar
- Takahashi, M., Parallel Reductions in λ-Calculus, J. Symbolic Computation, 7, pp. 113–123, 1989.Google Scholar
- Tatsuta, M., Program Synthesis Using Realizability, Theoret. Computer Sci., to appear.Google Scholar