A criterion for the separable axiomatization of Gödel's $S_n $
Tsutomu Hosoi · Proceedings of the Japan Academy Series A Mathematical Sciences · 1967
This report is an extension to our papers E2] and E3].And we use notations and results of them without mentioning.In this paper, we report a criterion for an axiom scheme togive a separable axiomatic system for S by adding it to Dummett's LC, and we also report that there is no intermediate axiomatic system between S and S+.Our result is also an extension to that obtained by Hanazawa [1 in the following form, though we do not suppose familiarity with it.Theorem 1 (By Hanazawa).The system LI/A is equivalent to the usual classical system $1 if and only if A is valid in $1 but not in $2.Since axiomatic systems are known for S's, the validity in S is equivalent to the provability in S. Though Hanazawa does not mention explicitly, the above theorem implies the following Corollary 2. There is not an intermediate axiomatic system between SI and S2.