Conservativity of Type Theory over Higher-order Arithmetic

TL;DR

This paper explores the conservativity of various dependent type theories over higher-order arithmetic, showing how different interpretations impact their proof capabilities and extending classical results to richer type theories.

Contribution

It extends classical conservativity results to type theories with universes and analyzes the influence of logical interpretations on their proof strength.

Findings

Type theories with one universe are conservative over Higher-order Heyting Arithmetic.

Proof-irrelevant interpretations prove the same higher-order arithmetical formulas as HAH.

Proof-relevant interpretations prove more second-order formulas but the same first-order formulas as HAH.

Abstract

We investigate how much type theory is able to prove about the natural numbers. A classical result in this area shows that dependent type theory without any universes is conservative over Heyting Arithmetic (HA). We build on this result by showing that type theories with one level of universes are conservative over Higher-order Heyting Arithmetic (HAH). Although this clearly depends on the specific type theory, we show that the interpretation of logic also plays a major role. For proof-irrelevant interpretations, we will see that strong versions of type theory prove exactly the same higher-order arithmetical formulas as HAH. Conversely, proof-relevant interpretations prove strictly more second-order arithmetical formulas than HAH, however they still prove exactly the same first-order arithmetical formulas. Along the way, we investigate the different interpretations of logic in type theory, and to what extent dependent type theories can be seen as extensions of higher-order logic. We apply our results by proving De Jongh's theorem for type theory.