A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π1-equivalent to KP

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

“…Especially, the reducibility of Σ 1 1 -AC 0 to AETJ +(T-I N ) was already given by Feferman from the beginning of explicit mathematics, but it was via the reducibility of Σ 1 1 -AC 0 to ACA 0 which does not tell how to simulate the axiom of choice (especially those inside cut-non-free proofs) within ACA 0 or AETJ + (T-I N ). With the method developed by the present author and Zumbrunnen [37], we will give interpretations of subsystems of second order arithmetic in the systems of explicit mathematics, and this automatically tells how to simulate the axiom of choice in explicit mathematics. As expected, for this simulation, the join operator j of explicit mathematics will play the crucial role.…”

“…The method developed by the present author and Zumbrunnen [37] is to interpret classical theories in classical theories but via non-classical intuitionistic theories. Quite interestingly, in spite of this intuitionistic nature, this method does not work for intuitionistic theories so well, as explained in [37,Subsection A.4] (and this is why we have devoted one subsection for the argument on the logic on which explicit mathematics should be formulated).…”

“…In the (modified) sense of Visser's miniature model theory, this can be seen as the third kind of model construction method for classical theories, after Cohen's forcing method and Krivine's classical realizability method. Whereas the present author and Zumbrunnen [37] have applied this method in order to give lower bounds to a new family of theories introduced there, here in the present paper we apply the method to Brouwer's analysis based on f.c.s. [free choice sequence] has been studied in various logical formalisms by Kleene, Vesley, Kreisel, Troelstra, van Dalen and others (cf.…”

“…The present author and Zumbrunnen [37] have developed a new method to construct an interpretation between classical theories (i.e., theories based on classical logic) by composing interpretations between intuitionistic theories (theories based on intuitionistic logic, which of course include classical theories). In the (modified) sense of Visser's miniature model theory, this can be seen as the third kind of model construction method for classical theories, after Cohen's forcing method and Krivine's classical realizability method.…”

“…Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinal analysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen [37], which can be seen as the third kind of model construction method for classical theories, after Cohen's forcing and Krivine's classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories.…”