nLab
continuum hypothesis

Idea

The continuum hypothesis is a statement of set theory which says, roughly, that every set 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.

Statement

Definition

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

$$N\hookrightarrow B\hookrightarrow {\Omega }^N$$N \hookrightarrow B\hookrightarrow \Omega^N

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

$$\mid \mathbb{N}\mid <\alpha <\mid {\Omega }^{\mathbb{N}}\mid$${|\mathbb{N}|} \lt \alpha\lt {|\Omega^\mathbb{N}|}

which it is more common to write as

$${\aleph }_0<\alpha <2^{{\aleph }_0}$$\aleph_0 \lt \alpha \lt 2^{\aleph_0}

Unprovability

Theorem

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 $\mathrm{Set}$ is $2≔\{0,1\}$. The technique of constructing such a topos is called forcing.

Definition

(Cohen poset)

Let $\mathbb{N}$ be the set of natural numbers; i.e. the natural-numbers object in $\mathrm{Set}$. Let $B$ be a set with strictly larger cardinality $\mid B\mid >\mid \mathbb{N}\mid$; 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

$$p:F_p\to 2$$p:F_p\to 2

where $F_p\subseteq B\times \mathbb{N}$ is any finite subset. The order relation on $P$ is defined by

$$q\le p\mathrm{iff}{F}_q\supseteq F_p\mathrm{and}q{\mid }_{F_p}=p$$q\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p

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

$$f:B\to 2^{\mathbb{N}}$$f:B\to 2^\mathbb{N}

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.

Lemma

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

Lemma

Let $k_{B\times \mathbb{N}}:\{\begin{array}{l}P\to \mathrm{Set}\\ p\mapsto B\times \mathbb{N}\end{array}$ denote the functor constant on $B\times \mathbb{N}$. Let

$$A:\{\begin{array}{l}P\to \mathrm{Set}\\ p\mapsto \{(b,n)\mid p(b,n)=0\}\subseteq B\times \mathbb{N}\end{array}$$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 $\mathrm{Sub}(k_{B\times \mathbb{N}})$; 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 $\mathrm{Psh}(P)$. Let ${\Omega }_{\neg \neg }$ denote the subobject classifier of $\mathrm{Sh}(P,\neg \neg )$. Recall that ${\Omega }_{\neg \neg }$ is given by the equalizer ${\Omega }_{\neg \neg }=\mathrm{eq}({\mathrm{id}}_{\Omega },\neg \neg )$.

By the preceding lemma, 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 $f:k_{B\times \mathbb{N}}\to {\Omega }_{\neg \neg }$.

Lemma

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.

Unrefutability

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 $V$.

References

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in geometry and logic, VI.2, VI.3

• M.C. Fitting, Intuitionistic logic, model theory and forcing, North-Holland (1969)