nLab
continuum hypothesis

The Continuum Hypothesis

Idea

Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space? In other terms, the question is: How many different sets of integers do there exist? K. Gödel (1947, p.515)

The continuum hypothesis is a famous problem of set theory concerning the cardinality of the real numbers (the “continuum”). The hypothesis in its classical form goes back to G. Cantor and was on top of Hilbert's millenium list of open problems in mathematics in 1900.

In concise form the continuum hypothesis (CHCH) reads: 2 0= 1\quad 2^{\aleph_0}=\aleph _1\quad; which roughly says that every subset of the real numbers is either countable or has the same cardinality as the set of all real numbers.

The generalized continuum hypothesis (GCHGCH) states more generally: 2 k= k+1\quad 2^{\aleph_k}=\aleph _{k+1}\quad.

The independence of the continuum hypothesis from the ZFC axioms of set theory has been established in landmark papers by K. Gödel and P. J. Cohen, the former proving the consistency of ZFC+CHZFC+CH relative to ZFCZFC in 1938, and the latter proving the consistency of ZFC+¬CHZFC+\neg CH relative to ZFCZFC in 1963.

The broader implications of the independence results for set theory in general and ZFCZFC in particular are somewhat controversial. They are widely taken as a pointer towards the deficiency of ZFCZFC and the need for further axioms of set theory.

W. Lawvere in 2003 interpreted Cantor’s original point of view as saying that CHCH holds for ‘sufficiently structureless’ sets and, accordingly, viewed Gödel’s 1938 result as a proof of CHCH, whereas in P. Dehornoy’s 2003 reinterpretation based on work of Woodin, CHCH is actually conjectured to be false.

The attempt to give categorical accounts of the forcing methods introduced by Cohen provided a strong impetus in the development of (elementary) topos theory in the work of Freyd, Tierney, Lawvere and later Scedrov. The following exposition follows this categorical approach.

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

  • J. L. Bell, Set Theory - Boolean-Valued Models and Independence Proofs , Oxford Logic Guides 47 3rd ed. Oxford UP 2005.

  • A. Church, Paul J. Cohen and the Continuum Problem, pp.15-20 in Proceedings ICM Moscow 1966. (pdf)

  • P. J. Cohen, The independence of the continuum hypothesis I, Proc.Nat.Acad.Sci. 50 (1963) pp.1143-1148. (pdf)

  • P. J. Cohen, The independence of the continuum hypothesis II, Proc.Nat.Acad.Sci. 51 (1963) pp.105-110. (pdf)

  • P. J. Cohen, Set Theory and the Continuum Hypothesis , Benjamin New York 1966. (Dover reprint 2008)

  • P. Dehornoy, Progrès récents sur l’hypothèse du continu (d’après Woodin) , Séminaire Bourbaki exposé 915 (2003). (English version)

  • M.C. Fitting, Intuitionistic Logic, Model Theory and Forcing, North-Holland Amsterdam 1969.

  • K. Gödel, What is Cantor’s continuum problem? , Am. Math. Monthly 54 no. 9 (1947) pp.515-25. (pdf)

  • F. W. Lawvere, Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic 9 no.2 (2003) pp.213-224. (ps-preprint)

  • Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (sections VI.2, VI.3)

Revised on August 17, 2014 01:57:22 by Thomas Holder (82.113.121.67)