The continuum hypothesis is a statement of set theory which says, roughly, that every subset of real numbers is either countable or has the same cardinality as the set of all real numbers (“the continuum”). It cannot be proven or disproven from any of the usual axioms of set theory.
then either the first or the second is an isomorphism.
which it is more common to write as
One topos for which the theorem holds is called the Cohen topos; it is the topos of sheaves with respect to the dense topology? (also called the -topology) on the Cohen poset. Thus, in this topos, there exist monomorphisms that are both not isomorphisms.
Let be the set of natural numbers; i.e. the natural-numbers object in . Let be a set with strictly larger cardinality ; e.g. will do because of the diagonal argument. Then the Cohen poset is defined to be the set of morphisms
where is any finite subset. The order relation on is defined by
where the right-hand condition means that restricted to must coincide with .
We think of each element of as an approximation to the function that is the transpose of the putative monomorphism
with “smaller” elements considered as better approximations. The very rough intuition is that (if ) forms a codirected diagram of monomorphisms with domains of increasing size whose colimit is , and that by free cocompletion (i.e. forming (pre)sheaves) we obtain a topos in which this colimit exists.
Let denote the functor constant on . Let
Then we have in ; i.e. is a closed subobject with respect to the dense topology in the algebra of subobjects of .
Let denote the subobject classifier of . Let denote the subobject classifier of . Recall that is given by the equalizer .
By the preceding lemma, the characteristic morphism of the subobject factors through some .
The adjoint of is a monomorphism.
The associated-sheaf functor sends to a monomorphism in the Cohen topos.
Stanford Encyclopedia of Philosophy, The Continuum Hypothesis
M.C. Fitting, Intuitionistic logic, model theory and forcing, North-Holland (1969)