Church's thesis (CT) as an axiom in constructive logic states that every total function of type $\mathbbN \to \mathbbN$ is computable, i.e. definable in a model of computation. CT is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak König's Lemma and the fan theorem, respectively. Recently, CT was proved consistent for (univalent) constructive type theory.
Since neither Weak König's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that CT is only inconsistent with a combination of classical logic and choice axioms. We study consequences of CT and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms.
We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.