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

Abstract

We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman's system T 0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretation of the subsystem Σ 1 2 -AC + (BI) of second order arithmetic inside T 0 . 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.

A Marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic

“…Whereas the first two correspond to unique and non-unique axioms of choice, respectively, the others seem specific to set theory. As we want to have and to stay within the strength of PRA, we shall consider "weak weak" set theory in the sense of the second author [37].…”

Proof Theory of Constructive Systems: Inductive Types and Univalence

“…For a long time [34] was also the only proof that enabled one to reduce the classical theories (∆ 1 2 -CA) + BI and KPi to classical T 0 . There is now also a proof by Sato [65] for the reductions in the classical case that avoids proof-theoretic ordinals. However, determining the strength of other important fragments of MLTT (such as the ones analyzed by Setzer in [66]) still requires the techniques of ordinal analysis.…”