Classical Predicative Logic-Enriched Type Theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl’s foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.

💡 Research Summary

The paper presents a detailed study of how classical predicative systems of second‑order arithmetic can be faithfully represented within the framework of logic‑enriched type theories (LTTs). The authors begin by recalling LTTW, a logic‑enriched type theory designed to formalise Hermann Weyl’s predicative foundation as presented in Das Kontinuum. LTTW extends a simple dependent type theory with a primitive layer for forming and proving propositions, thereby allowing both type‑theoretic constructions and classical logical reasoning to coexist in a single formal system.

Within LTTW the authors isolate two subsystems, named LTTO and LTTO*. Both are deliberately constrained to mirror two well‑known subsystems of second‑order arithmetic: ACA₀ (arithmetical comprehension with a predicative restriction) and ACA (full arithmetical comprehension). LTTO enforces a strict predicative discipline: any set‑forming operation may only refer to objects that have already been constructed at lower type levels. This mirrors the “predicative” comprehension axiom of ACA₀. LTTO* relaxes this restriction, allowing comprehension over any arithmetical predicate, and therefore corresponds exactly to ACA. In both systems the underlying type language includes natural numbers, function types, and a universe of sets, while the logical layer is classical first‑order logic enriched with second‑order quantifiers.

The technical heart of the paper is a novel conservativity‑preserving translation technique. Rather than using model‑theoretic arguments, the authors define an explicit syntactic interpretation of the stronger system (LTTW) inside the weaker one (LTTO). This interpretation proceeds by mapping each term, type, and proof of LTTW to a term, type, and proof of LTTO in a compositional way. Crucially, the mapping respects the predicative hierarchy: higher‑order set‑forming expressions are encoded as lower‑level constructions that are provably well‑typed and preserve the original meaning. The authors prove two lemmas: (1) type preservation – the translation of a well‑typed LTTW expression is well‑typed in LTTO, and (2) proof preservation – if a proposition is provable in LTTW, its translation is provable in LTTO. Together these lemmas establish that LTTW is a conservative extension of LTTO; consequently, any theorem of ACA₀ that can be expressed in LTTW is already provable in LTTO.

Having secured the conservativity result, the authors turn to the reverse direction: they give explicit translations from ACA₀ to LTTO and from ACA to LTTO*. These translations are straightforward: second‑order variables become variables of the set‑type, arithmetical formulas are mapped verbatim, and comprehension axioms are rendered as type‑theoretic set‑formation rules that satisfy the predicative constraints of the target LTT. The authors verify that the translations are sound and conservative, i.e., no new first‑order arithmetic facts are introduced, and that the image of each system under translation is exactly the set of theorems of the corresponding LTT.

Putting the pieces together, the paper demonstrates three main results: (i) LTTO is proof‑theoretically equivalent to ACA₀, (ii) LTTO* is proof‑theoretically equivalent to ACA, and (iii) LTTW strictly extends LTTO (and therefore ACA₀) while being conservative over LTTO*. In other words, LTTW can prove all theorems of ACA and more, confirming the long‑standing claim that Weyl’s predicative foundation is stronger than ACA₀.

Beyond these specific comparisons, the authors argue that the syntactic interpretation technique they introduced is of independent interest. Because it works by constructing an explicit translation of expressions rather than appealing to semantic models, it can be adapted to a wide variety of settings: other logic‑enriched type theories, intuitionistic or constructive foundations, and even higher‑order systems where traditional model‑theoretic conservativity proofs are difficult. The paper concludes with suggestions for future work, including extending the method to impredicative type theories, investigating computational interpretations of the translations, and exploring connections with proof‑theoretic ordinal analysis. Overall, the work provides a clear bridge between classical predicative arithmetic and modern type‑theoretic frameworks, and it supplies a versatile tool for establishing relative strength results among enriched type theories.