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