For context see [[the topos Set]]. +-- {: .num_defn} ###### Definition The *continuum hypothesis* asserts that there is no strict inequality of [[cardinal numbers]] $$|\mathbb{N}|\lt \alpha\lt |\Omega^\mathbb{N}|$$ where the leftest symbol dnotes the cardinality of the [[natural-numbers object]] $\mathbb{N}$ in [[Set]] and the rightest symbol denotes its [[power object]]. =-- +-- {: .num_theorem} ###### Theorem There exists a boolean topos in which the axiom of choice holds and the continuum hypothesis fails. =-- +-- {: .num_defn} ###### Definition ([[Cohen topos]]) =-- ## References * André Joyal, Ieke Moerdijk, sheaves in geometry and logic, VI.2, VI.3 * M.C. Fitting, "Intuitionistic logic, model theory and forcing" , North-Holland (1969)