+-- {: .num_defn} ###### Definition The *continuum hypothesis* asserts that there is no strict inequality of [[cardinal numbers]] $$|\mathbb{N}|\lt \alpha\lt |\mathbf{2}^\mathbb{N}|$$ =-- +-- {: .num_theorem} ###### Theorem There exists a boolean topos in which the axiom of choice holds and the continuum hypothesis fails. =-- ## References André Joyal, Ieke Moerdijk, sheaves in geometry and logic, VI.2, VI.3