Universes in explicit mathematics

An Extended Predicative Definition of the Mahlo Universe

“…EETJ adds types on the second order level, and axiomatize elementary comprehension and join as type construction operations. We dispense here with a detailed description of EETJ which can be found in many papers on Explicit Mathematics (e.g., [JKS01], [JS02] or [Kah07]). Let us just briefly address the finite axiomatization of elementary comprehension and join.…”

“…We now turn to the notion of universes as discussed, for instance, in [JKS01]. In the context of Mahloness, universes are considered by JÄGER, STRAHM, and STUDER [JS01, JS02, Str02, Jäg05, JS05].…”

“…A detailed discussion of the concept of universes in Explicit Mathematics can be found in [JKS01], including least universes and name induction. Universes can be considered as a formalization of admissibility.…”

“…For instance, in [JKS01], a concept of name strictness is introduced. It expresses that generators only generate names for appropriate arguments (e.g.,…”

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

“…On this line, some extensions by small large cardinal notions have been introduced and investigated in Marzetta [28], Strahm [39,40], Jäger, Kahle and Studer [19], Jäger and Studer [24], Jäger and Strahm [21,23] and Jäger [18], as similar extensions of Kripke-Platek set theory, of constructive Zermelo-Fraenkel set theory and of Martin-Löf type theory have been considered in the right direction to strengthen these mathematical frameworks and also to develop further proof-theoretic investigations. Also for the extensions by monotone inductive definition, another direction of development, Takahashi [41], Rathjen [30][31][32]35] and Glaß, Rathjen and Schlüter [14] all formulated the systems on classical logic, though Rathjen [32] mentioned extending the result for the intuitionistic version as a further problem which was treated by Tupailo [46].…”