Classical Predicative Logic-Enriched Type Theories

TL;DR

This paper constructs two logic-enriched type theories that closely correspond to classical predicative systems of second order arithmetic, and demonstrates their conservativity and relative strength, advancing formal foundations of mathematics.

Contribution

It introduces two new LTTs, LTTO and LTTO*, that model classical predicative systems and provides a novel technique for proving conservativity between LTTs.

Findings

LTTO and LTTO* correspond to ACAO and ACA systems

LTTW is strictly stronger than ACAO

A new technique for proving conservativity of LTTs

Abstract

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.