Conservativity of Type Theory over Higher-order Arithmetic
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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.
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