Spahn continuum hypothesis

For context see the topos Set?.

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?.

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 $2:=\{0,1\}$.

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 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.

Lemma

The dense? Lawvere-Tierney topology? on $Psh(P)$ is subcanonical. In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(p,\neg\neg)$

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 $k_{B\times\mathbb{N}}$.

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.

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)