basic constructions:
strong axioms
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 ($CH$) reads: $\quad 2^{\aleph_0}=\aleph _1\quad$; 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 ($GCH$) states more generally: $\quad 2^{\aleph_k}=\aleph _{k+1}\quad$.
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 $ZFC+CH$ relative to $ZFC$ in 1938, and the latter proving the consistency of $ZFC+\neg CH$ relative to $ZFC$ in 1963.
The broader implications of the independence results for set theory in general and $ZFC$ in particular are somewhat controversial. They are widely taken as a pointer towards the deficiency of $ZFC$ and the need for further axioms of set theory.
W. Lawvere in 2003 interpreted Cantor’s original point of view as saying that $CH$ holds for ‘sufficiently structureless’ sets and, accordingly, viewed Gödel’s 1938 result as a proof of $CH$, whereas in P. Dehornoy’s 2003 reinterpretation based on work of Woodin, $CH$ 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.
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
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
which it is more common to write as
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 $Set$ is $2\coloneqq \{0,1\}$. The technique of constructing such a topos is called forcing.
(Cohen poset)
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. $B\coloneqq 2^{2^{\mathbb{N}}}$ will do because of the diagonal argument. Then the Cohen poset $P$ is defined to be the set of morphisms
where $F_p\subseteq B\times \mathbb{N}$ is any finite subset. The order relation on $P$ is defined by
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
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.
The dense? Grothendieck topology on $P$ is subcanonical. In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(p,\neg\neg)$
Let $k_{B\times\mathbb{N}}:\begin{cases}P\to Set \\ p \mapsto B\times\mathbb{N}\end{cases}$ denote the functor constant on $B\times\mathbb{N}$. Let
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 $k_{B\times\mathbb{N}}$.
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 given by the equalizer $\Omega_{\neg\neg}=eq(id_\Omega,\neg\neg)$.
By the preceding lemma, the characteristic morphism $\chi_a$ of the subobject $a \colon A\hookrightarrow k_{B\times\mathbb{N}}=k_B\times\k_\mathbb{N}$ factors through some $f \colon k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}$.
The adjoint $g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}}$ of $f$ is a monomorphism.
The associated-sheaf functor sends $g$ to a monomorphism in the Cohen topos.
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 $V$.
Just how flexible can the power operation $\kappa \mapsto 2^\kappa$ be? There are of course some constraints. Obvious ones are that $\kappa \lt 2^\kappa$ and $2^\kappa \leq 2^\lambda$ whenever $\kappa \leq \lambda$. A more refined one is a consequence of König’s theorem, namely that
where the right side is the cofinality of $2^\kappa$.
A remarkable illustration of the power of the forcing method is Easton’s theorem, which says that as far as regular cardinals go, these are really the only constraints.
(Easton) Suppose $\mathcal{M}$ is a model of ZFC in which the generalized continuum hypothesis (GCH) holds. Let $F$ be a partial function from the class of infinite regular cardinals to the class of cardinals such that
$F$ preserves the order $\leq$;
$\kappa$ is less than the cofinality of $F(\kappa)$ for all $\kappa \in dom(F)$.
Then there is a generic extension $\mathcal{M}[G]$ of $\mathcal{M}$ with the same cardinals and cofinalities, such that $\mathcal{M}[G] \models 2^\kappa = F(\kappa)$ for all $\kappa \in dom(F)$.
On the other hand, the behavior of the power operation on singular cardinals is not so unconstrained. For example, in a model of ZFC, the smallest cardinal for which GCH fails can never be singular. The so-called “pcf theory?” (or “possible cofinalities theory”), due to Saharon Shelah, gives some information on possible bounds for the power operation on singular cardinals (among other things).
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)