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\section*{continuum hypothesis}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{the_continuum_hypothesis}{}\section*{{The Continuum Hypothesis}}\label{the_continuum_hypothesis}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{unprovability}{Unprovability}\dotfill \pageref*{unprovability} \linebreak
\noindent\hyperlink{unrefutability}{Unrefutability}\dotfill \pageref*{unrefutability} \linebreak
\noindent\hyperlink{EastonTheorem}{Generalization: Easton's theorem}\dotfill \pageref*{EastonTheorem} \linebreak
\noindent\hyperlink{links}{Links}\dotfill \pageref*{links} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\begin{quote}%
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. G\"o{}del (\hyperlink{Goedel47}{1947}, p.515)


\end{quote}
The \textbf{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 [[Georg Cantor|G. Cantor]] and was on top of [[Hilbert's problems|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 set|countable]] or has the same [[cardinality]] as the set of all real numbers.

The \emph{generalized continuum hypothesis} ($GCH$) states more generally: $\quad 2^{\aleph_k}=\aleph _{k+1}\quad$. (But see also Remark \ref{GCHinZF} below.)

The \textbf{independence of the continuum hypothesis} from the [[ZFC]] axioms of set theory has been established in landmark papers by [[Kurt Gödel|K. Gödel]] and [[Paul J. Cohen|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. For fully [[formal proof]] see \hyperlink{Han18}{Han 18}.

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 \emph{new axioms} of set theory. This position has been voiced famously in \hyperlink{Goedel47}{G\"o{}del (1947)} from a [[platonism|platonist perspective]].

[[William Lawvere|W. Lawvere]] in \hyperlink{Lawvere03}{2003} interpreted Cantor's original point of view as saying that $CH$ holds for `sufficiently structureless' sets and, accordingly, viewed G\"o{}del's 1938 result as a proof of $CH$, whereas in [[Patrick Dehornoy|P. Dehornoy]]`s 2003 reinterpretation based on work of \emph{Woodin}, $CH$ is actually conjectured to be false. [[Solomon Feferman|S. Feferman]] has argued more recently that $CH$ is essentially \emph{mathematically indefinite} and has made notions of `indefiniteness' explicit that indeed enable to back this point of view with technical results (cf. \hyperlink{Feferman11}{Feferman (2011)}). Similarly, [[Joel David Hamkins]] has argued (cf. \hyperlink{Hamkins12}{Hamkins 2012}) that under the [[set-theoretic multiverse|multiverse]] view of set theory, $CH$ is settled by our extensive knowledge of models of set theory satisfying both $CH$ and $\neg CH$ which all seem fully set-theoretic, so that no principle implying either one can possibly be considered `obviously true'.

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 [[Peter Freyd|Freyd]], [[Miles Tierney|Tierney]], Lawvere and later Scedrov. The following exposition follows this categorical approach.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{defn}
\label{}\hypertarget{}{}
Let $E$ be an [[elementary topos]] with [[subobject classifier]] $\Omega$ and [[natural numbers object]] $N$. The (external) \textbf{continuum hypothesis} in $E$ asserts that if there is a sequence of [[monomorphisms]]

\begin{displaymath}
N \hookrightarrow B\hookrightarrow \Omega^N
\end{displaymath}
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]]

\begin{displaymath}
{|\mathbb{N}|} \lt \alpha\lt {|\Omega^\mathbb{N}|}
\end{displaymath}
which it is more common to write as

\begin{displaymath}
\aleph_0 \lt \alpha \lt 2^{\aleph_0}
\end{displaymath}
\end{defn}
\hypertarget{unprovability}{}\subsection*{{Unprovability}}\label{unprovability}

\begin{theorem}
\label{}\hypertarget{}{}
There exists a [[boolean topos]] in which the [[axiom of choice]] holds and the continuum hypothesis fails.

\end{theorem}
One topos for which the theorem holds is called the \emph{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]].

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(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 \emph{Cohen poset} $P$ is defined to be the set of morphisms

\begin{displaymath}
p:F_p\to 2
\end{displaymath}
where $F_p\subseteq B\times \mathbb{N}$ is any [[finite set|finite]] subset. The order relation on $P$ is defined by

\begin{displaymath}
q\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p
\end{displaymath}
where the right-hand condition means that $q$ restricted to $F_p$ must coincide with $p$.

\end{defn}
We think of each element of $P$ as an approximation to the function $F:B\times\mathbb{N}$ that is the [[exponential|transpose]] of the putative monomorphism

\begin{displaymath}
f:B\to 2^\mathbb{N}
\end{displaymath}
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.

\begin{lemma}
\label{}\hypertarget{}{}
The [[dense topology|dense]] [[Grothendieck topology]] on $P$ is [[subcanonical topology|subcanonical]]. In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(P,\neg\neg)$

\end{lemma}
\begin{lemma}
\label{}\hypertarget{}{}
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

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

\end{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 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 $f \colon k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}$.

\begin{lemma}
\label{}\hypertarget{}{}
The adjoint $g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}}$ of $f$ is a monomorphism.

\end{lemma}
\begin{corollary}
\label{}\hypertarget{}{}
The associated-sheaf functor sends $g$ to a monomorphism in the Cohen topos.

\end{corollary}
\hypertarget{unrefutability}{}\subsection*{{Unrefutability}}\label{unrefutability}

If $V$ is a model of [[ZF]], then the continuum hypothesis and the [[axiom of choice]] both hold in G\"o{}del's [[constructible universe]] $L$ built from $V$. Actually, the GCH holds in $L$ as well.

\begin{remark}
\label{GCHinZF}\hypertarget{GCHinZF}{}
Regarding the statement of the \emph{generalized} continuum hypothesis in \emph{ZF} (not ZFC), one should distinguish various possibilities. One might leave the statement $2^{\aleph_n} = \aleph_{n+1}$ unchanged, so that the GCH becomes a statement just about ordinals or well-ordered sets. But then one could argue such a generalized continuum hypothesis is not as general or strong as it might be, since not all sets can be well-ordered using ZF alone. The more general statement would say that if there are monomorphisms $X \to Y$ and $Y \to P(X)$, then $Y$ is bijective with one of $X, P(X)$.

For example, Sierpiski proved that over ZF, the generalized continuum hypothesis implies AC. (See [[Hartogs number]].) For this result, he certainly used the stronger formulation.

\end{remark}
\hypertarget{EastonTheorem}{}\subsection*{{Generalization: Easton's theorem}}\label{EastonTheorem}

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

\begin{itemize}%
\item $\kappa \lt cof(2^\kappa)$

\end{itemize}
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 \emph{only} constraints.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(\hyperlink{Easton70}{Easton 70})}

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

\begin{itemize}%
\item $F$ is strictly increasing and preserves the order $\leq$;


\item $\kappa$ is less than the cofinality of $F(\kappa)$ for all $\kappa \in dom(F)$.



\end{itemize}
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 $\kappa \in dom(F)$.

\end{theorem}
On the other hand, the behavior of the power operation on [[regular cardinal|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).

\hypertarget{links}{}\subsection*{{Links}}\label{links}

\begin{itemize}%
\item Stanford Encyclopedia of Philosophy, \emph{\href{http://plato.stanford.edu/entries/continuum-hypothesis/}{The Continuum Hypothesis}}


\item MO \emph{Solutions to the Continuum Hypothesis} . (\href{http://mathoverflow.net/questions/23829/solutions-to-the-continuum-hypothesis}{link})



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John Bell|J. L. Bell]], \emph{Set Theory - Boolean-Valued Models and Independence Proofs} , Oxford Logic Guides \textbf{47} 3rd ed. Oxford UP 2005.


\item [[Alonzo Church|A. Church]], \emph{Paul J. Cohen and the Continuum Problem}, pp.15-20 in Proceedings ICM Moscow 1966. (\href{http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0015.0020.ocr.pdf}{pdf})


\item [[Paul J. Cohen|P. J. Cohen]], \emph{The independence of the continuum hypothesis I}, Proc. Nat. Acad. Sci. \textbf{50} (1963) pp.1143-1148. (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287/pdf/pnas00240-0135.pdf}{pdf})


\item [[Paul J. Cohen|P. J. Cohen]], \emph{The independence of the continuum hypothesis II}, Proc. Nat. Acad. Sci. \textbf{51} (1963) pp.105-110. (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611/pdf/pnas00175-0117.pdf}{pdf})


\item [[Paul J. Cohen|P. J. Cohen]], \emph{Set Theory and the Continuum Hypothesis} , Benjamin New York 1966. (Dover reprint 2008)


\item [[Patrick Dehornoy|P. Dehornoy]], \emph{Progr\`e{}s r\'e{}cents sur l'hypoth\`e{}se du continu (d'apr\`e{}s Woodin)} , S\'e{}minaire Bourbaki expos\'e{} \textbf{915} (2003). (\href{http://www.math.unicaen.fr/~dehornoy/Surveys/DgtUS.pdf}{English version})


\item [[Solomon Feferman]], \emph{The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem} , Harvard lectures 2011. (\href{http://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf}{pdf})


\item [[Joel David Hamkins]], \emph{The set-theoretic multiverse}, Review of Symbolic Logic 5:416-449 (2012), \href{https://arxiv.org/abs/1108.4223}{arxiv}


\item M.C. Fitting, \emph{Intuitionistic Logic, Model Theory and Forcing}, North-Holland Amsterdam 1969.


\item [[K. Gödel]], \emph{What is Cantor's continuum problem?} , Am. Math. Monthly \textbf{54} no. 9 (1947) pp.515-25. (\href{http://www.personal.psu.edu/ecb5/Courses/M475W/Readings/Week06-Sep30-IntoTheTwentiethCentury/01-WhatisCantorsContinuumProblembyKurtGodel.pdf}{pdf})


\item [[William Lawvere|F. W. Lawvere]], \emph{Foundations and Applications: Axiomatization and Education}, Bulletin of Symbolic Logic \textbf{9} no.2 (2003) pp.213-224. (\href{https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps}{ps-preprint})


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994. (sections VI.2, VI.3)


\item [[W. Hugh Woodin]], \emph{The Continuum Hypothesis, Part I} , Notices AMS \textbf{48} no.6 (2001) pp.567-576. (\href{http://www.ams.org/notices/200106/fea-woodin.pdf}{pdf})


\item [[W. Hugh Woodin]], \emph{The Continuum Hypothesis, Part II} , Notices AMS \textbf{48} no.7 (2001) pp.681-690. (\href{http://www.ams.org/notices/200107/fea-woodin.pdf}{pdf})


\item W. Easton, \emph{Powers of regular cardinals}, Ann. Math. Logic, 1 (2): 139--178, (1970) \href{https://dx.doi.org/10.1016%2F0003-4843%2870%2990012-4}{doi:10.1016/0003-4843(70)90012-4}



\end{itemize}
A [[formal proof]] of the independence of the continuum hypothesis from [[ZFC]] is in

\begin{itemize}%
\item [[Jesse Han]], \emph{Flypitch project -- Formal proof of the independence of CH} (\href{https://github.com/flypitch}{github:flypitch}, \href{https://github.com/flypitch/flypitch-notes/blob/master/forcing-notes.pdf}{pdf})

\end{itemize}
[[!redirects CH]] [[!redirects Continuum hypothesis]] [[!redirects GCH]] [[!redirects generalized continuum hypothesis]]

[[!redirects Easton's theorem]] [[!redirects Easton theorem]]



\end{document}