basic constructions:
strong axioms
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.
Let $E$ be an elementary topos with subobject classifier $\Omega$ and natural numbers object $N$. The (external) continuum hypothesis in $E$ asserts that if there is a sequence of monomorphisms
then either the first or the second is an isomorphism.
In the classical case (that is, in the topos Set with the axiom of choice), this equivalently asserts that there is no strict inequality of cardinal numbers
which it is more common to write as
There exists a boolean topos in which the axiom of choice holds and the continuum hypothesis fails.
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 $\neg\neg$-topology) on the Cohen poset. Thus, in this topos, there exist monomorphisms $\mathbb{N} \hookrightarrow B\hookrightarrow 2^{\mathbb{N}}$ that are both not isomorphisms.
The Cohen topos will be constructed from the topos Set of sets. For this, recall that the subobject classifier of $Set$ is $2\coloneqq \{0,1\}$. The technique of constructing such a topos is called forcing.
(Cohen poset)
Let $\mathbb{N}$ be the set of natural numbers; i.e. the natural-numbers object in $Set$. Let $B$ be a set with strictly larger cardinality ${|B|}\gt {|\mathbb{N}|}$; e.g. $B\coloneqq 2^{2^{\mathbb{N}}}$ will do because of the diagonal argument. Then the Cohen poset $P$ is defined to be the set of morphisms
where $F_p\subseteq B\times \mathbb{N}$ is any finite subset. The order relation on $P$ is defined by
where the right-hand condition means that $q$ restricted to $F_p$ must coincide with $p$.
We think of each element of $P$ as an approximation to the function $F:B\times\mathbb{N}$ that is the transpose of the putative monomorphism
with “smaller” elements considered as better approximations. The very rough intuition is that $p\to q\to \dots$ (if $p\ge p\ge \dots$) forms a codirected diagram of monomorphisms with domains of increasing size whose colimit is $f$, and that by free cocompletion (i.e. forming (pre)sheaves) we obtain a topos in which this colimit exists.
The dense? Grothendieck topology on $P$ is subcanonical. In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(p,\neg\neg)$
Let $k_{B\times\mathbb{N}}:\begin{cases}P\to Set \\ p \mapsto B\times\mathbb{N}\end{cases}$ denote the functor constant on $B\times\mathbb{N}$. Let
Then we have $\neg\neg A=A$ in $Sub(k_{B\times\mathbb{N}})$; i.e. $A$ is a closed subobject with respect to the dense topology $\neg\neg$ in the algebra of subobjects of $k_{B\times\mathbb{N}}$.
Let $\Omega$ denote the subobject classifier of $Psh(P)$. Let $\Omega_{\neg\neg}$ denote the subobject classifier of $Sh(P,\neg\neg)$. Recall that $\Omega_{\neg\neg}$ is given by the equalizer $\Omega_{\neg\neg}=eq(id_\Omega,\neg\neg)$.
By the preceding lemma, the characteristic morphism $\chi_a$ of the subobject $a \colon A\hookrightarrow k_{B\times\mathbb{N}}=k_B\times\k_\mathbb{N}$ factors through some $f \colon k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}$.
The adjoint $g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}}$ of $f$ is a monomorphism.
The associated-sheaf functor sends $g$ to a monomorphism in the Cohen topos.
If $V$ is a model of ZF, then the continuum hypothesis and the axiom of choice both hold in Gödel’s constructible universe $L$ built from $V$.
Stanford Encyclopedia of Philosophy, The Continuum Hypothesis
Saunders Mac Lane, Ieke Moerdijk, sections VI.2, VI.3 of Sheaves in geometry and logic
M.C. Fitting, Intuitionistic logic, model theory and forcing, North-Holland (1969)