For context see [[the topos Set]]. +-- {: .num_defn} ###### Definition Let $E$ be an [[elementary topos]] with [[subobject classifier]] $\Omega$ and [[natural-numbers object]] $n$. The *continuum hypothesis* asserts that there is no sequence of monomorphisms $$n \hookrightarrow b\hookrightarrow \Omega^n$$ which are not isomorphisms. In the classical case this statement reads: 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. =-- The topos in which the theorem holds is called *Cohen topos*; it is the topos of sheaves with respect to the [[dense topology]] (also called $\neg\neg$-toology) on the Cohen poset. In this topos will exist a monomorphism $B\hookrightarrow 2^\mathbb{N}$ The cohen topos will be constructed from the topos [[Set]] of sets. for this recall that the subobject classifier of $Set$ is $2:=\{0,1\}$. +-- {: .num_defn} ###### Definition ([[Cohen topos]]) 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. $B:=2^{2^\mathbb{N}}$ will do because of the ''diagonal argument''. Then the *Cohen poset* $P$ is defined to be the set of morphisms $$p:F_p\to 2$$ where $F_p\subseteq B\times \mathbb{N}$ is a subset the order relation on $P$ is defined by $$q\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p$$ where the rightest condition means that $q$ restricted to $F_p$ shall coincide with $p$. We think of $P$ as a sequence of approximations to the function $F:B\times\mathbb{N}$ being the [[exponential|transpose]] of the putative monomorphism $$f:B\to 2^\mathbb{N}$$ and the smaller elements considered as the better approximations. The very rough intuition is that $p\to q\to \dots$ (if $p\ge p\ge \dots$) forms a [[codircted diagram]] of monomorphism with domains of increasing size whose colimit is $f$ and that by [[free cocompletion]] (here: forming (pre)sheaves) we obtain a topos in which this colimit exists. =-- +-- {: .num_lemma} ###### Lemma The [[dense topology|dense]] [[Lawvere-Tierney topology]] on $Psh(P)$ is subcanonical. In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(p,\neg\neg)$ =-- +-- {: .num_lemma} ###### Lemma Let $k_{B\times\mathbb{N}}:\begin{cases}P\to Set\\p\toB\times\mathbb{N}\end{cases}$ denote the functor constant on $B\times\mathbb{N}$. Let $$A:\begin{cases} P\to Set \\ p\mapsto \{(b,n)|p(b,n)=0\}\subseteq B\times \mathbb{N} \end{cases}$$ 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 $k_{B\times\mathbb{N}}$. =-- +-- {: .num_lemma} ###### Lemma 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 the equalizer $\Omega_{\neg\neg}=eq(id_\Omega,\neg\neg)$. The [[characteristic morphism]] $\chi_a$ of the subobject $a:A\hookrightarrow k_{B\times\mathbb{N}}=k_B\times\k_\mathbb{N}$ factors through some $f:k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}$. Then the adjoint $g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}}$ of $f$ is a monomorphism. =-- +-- {: .num_corollary} ###### Corollary The associated-sheaf functor sends $g$ to a monomorphism in the 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)