nLab
continuum hypothesis

The Continuum Hypothesis

Idea

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.

Statement

Definition

Let EE be an elementary topos with subobject classifier Ω\Omega and natural numbers object NN. The (external) continuum hypothesis in EE asserts that if there is a sequence of monomorphisms

NBΩ NN \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

||<α<|Ω |{|\mathbb{N}|} \lt \alpha\lt {|\Omega^\mathbb{N}|}

which it is more common to write as

0<α<2 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 B2 \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 SetSet is 2{0,1}2\coloneqq \{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 SetSet. Let BB be a set with strictly larger cardinality |B|>||{|B|}\gt {|\mathbb{N}|}; e.g. B2 2 B\coloneqq 2^{2^{\mathbb{N}}} will do because of the diagonal argument. Then the Cohen poset PP is defined to be the set of morphisms

p:F p2p:F_p\to 2

where F pB×F_p\subseteq B\times \mathbb{N} is any finite subset. The order relation on PP is defined by

qpiffF qF pandq| F p=pq\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p

where the right-hand condition means that qq restricted to F pF_p must coincide with pp.

We think of each element of PP as an approximation to the function F:B×F:B\times\mathbb{N} that is the transpose of the putative monomorphism

f:B2 f:B\to 2^\mathbb{N}

with “smaller” elements considered as better approximations. The very rough intuition is that pqp\to q\to \dots (if ppp\ge p\ge \dots) forms a codirected diagram of monomorphisms with domains of increasing size whose colimit is ff, 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 PP is subcanonical. In other words: For any pPp\in P we have y(p)=hom(,p)Sh(p,¬¬)y(p)=hom(-,p)\in\Sh(p,\neg\neg)

Lemma

Let k B×:{PSet pB×k_{B\times\mathbb{N}}:\begin{cases}P\to Set \\ p \mapsto B\times\mathbb{N}\end{cases} denote the functor constant on B×B\times\mathbb{N}. Let

A:{PSet p{(b,n)|p(b,n)=0}B×A:\begin{cases} P\to Set \\ p\mapsto \{(b,n)|p(b,n)=0\}\subseteq B\times \mathbb{N} \end{cases}

Then we have ¬¬A=A\neg\neg A=A in Sub(k B×)Sub(k_{B\times\mathbb{N}}); i.e. AA is a closed subobject with respect to the dense topology ¬¬\neg\neg in the algebra of subobjects of k B×k_{B\times\mathbb{N}}.

Let Ω\Omega denote the subobject classifier of Psh(P)Psh(P). Let Ω ¬¬\Omega_{\neg\neg} denote the subobject classifier of Sh(P,¬¬)Sh(P,\neg\neg). Recall that Ω ¬¬\Omega_{\neg\neg} is given by the equalizer Ω ¬¬=eq(id Ω,¬¬)\Omega_{\neg\neg}=eq(id_\Omega,\neg\neg).

By the preceding lemma, the characteristic morphism χ a\chi_a of the subobject a:Ak B×=k B×k a \colon A\hookrightarrow k_{B\times\mathbb{N}}=k_B\times\k_\mathbb{N} factors through some f:k B×Ω ¬¬f \colon k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}.

Lemma

The adjoint g:k BΩ ¬¬ k g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}} of ff is a monomorphism.

Corollary

The associated-sheaf functor sends gg to a monomorphism in the Cohen topos.

Unrefutability

If VV is a model of ZF, then the continuum hypothesis and the axiom of choice both hold in Gödel’s constructible universe LL built from VV.

References

Revised on March 10, 2014 10:19:14 by Urs Schreiber (82.113.121.82)