This thesis deals primarily with type-theoretic interpretations of constructive set theories using notions and ideas from homotopy type theory.
We introduce a family of interpretations [.]_k,h for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of the set theory BCS into the type theory H, in which sets and formulas are interpreted respectively as types of homotopy level k and h. Depending on the values of the parameters k and h we are able to interpret different theories, like Aczel's CZF and Myhill's CST. We relate the family [.]_k,h to the other interpretations of CST into homotopy type theory already studied in the literature in [UFP13] and [Gy16a].
We characterise a class of sentences valid in the interpretations [.]_k,∞ in terms of the ΠΣ axiom of choice, generalising the characterisation of [RT06] for Aczel's interpretation.
We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The formulas valid in this interpretation are then characterised in terms of the axiom of unique choice.
We extend the analysis of Aczel's interpretation provided in [GA06] to the interpretations of CST into homotopy type theory, providing a comparative analysis. This is done formulating in the logic-enriched type theory the key principles used in the proofs of the two interpretations.
We also investigate the notion of feasible ordinal formalised in the context of a linear type theory equipped with a type of resources. This type theory was originally introduced by Hofmann in [Hof03]. We disprove Hofmann's conjecture on the definable ordinals, by showing that for any given k ϵ N the ordinal ω^k is definable.
