On Church’s thesis in cubical assemblies

Abstract

We show that Church’s thesis, the axiom stating that all functions on the naturals are computable, does not hold in the cubical assemblies model of cubical type theory. We show that nevertheless Church’s thesis is consistent with univalent type theory by constructing a lex modality in cubical assemblies such that Church’s thesis holds in the corresponding reflective subuniverse.

“Upon This Quote I Will Build My Church Thesis”

“…As already hinted at before, this means that CT ∃ does not necessarily contradict function extensionality. Actually, we can even go much further: in the case where propositions are identified with hProps, CT ∃ turns out to be compatible not only with MLTT but also with full-blown univalence [35]. More generally and quite counterintuitively, univalence is compatible with many principles that would make the hardcore Bishop-style intuitionist raise a suspicious eyebrow, as long as they are squashed enough and thus made computationally harmless [33,35,37].…”

“…For starters, it immediately implies the existence of a quoting function and breaks both function extensionality and classical logic. The consistency of CT Σ with MLTT is an open problem that has been lingering for a while and seems to be considered a difficult question by experts [23,24,35]. The best result thus far [18] is the consistency of CT Σ with a stripped-down version of MLTT without the so-called 𝜉 rule:…”