Game semantics of Martin-L\"of type theory, part III: its consistency with Church's thesis

We prove consistency of intensional Martin-L"of type theory (MLTT) with formal Church’s thesis (CT), which was open for at least fifteen years. The difficulty in proving the consistency is that a standard method of realizability `{a} la Kleene does not work for the consistency, though it validates CT, as it does not model MLTT; specifically, the realizability does not validate MLTT’s congruence rule on pi-types (known as the $\xi$-rule). We overcome this point and prove the consistency by novel realizability `{a} la game semantics, which is based on the author’s previous work.