The independence of the continuum hypothesis. I, II.

*(English)*Zbl 0192.04401
Proc. Natl. Acad. Sci. USA 50, 1143-1148 (1963); 51, 105-110 (1964).

These two papers contain the solution of the famous Continuum Hypothesis. The author shows that the continuum hypothesis cannot be derived from the axioms of set theory. The proof is based an his method of forcing, which has become since very popular and led to numerous discoveries in the foundations of mathematics.

The author starts with an assumption that there is a countable transitive model \(M\) of Zermelo-Fraenkel set theory ZF which satisfies the axiom of constructibility and constructs an extension \(N\) of \(M\) which is a model of ZF + Axiom of Choice+\(2^{\aleph_0}>\aleph_1\). The idea is to obtain \(N\) by adjoining to \(M\) a sequence \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) of “generic” sets of integers, where \(\aleph_{\tau}\) is a cardinal number in \(M\), greater than \(\aleph_1\).

The key device is the notion of forcing. The forcing language consists of names of all elements of \(M\) and of generic sets \(a_\delta\), and of expressions using logical symbols and set-theoretical operations. A condition is a finite consistent set of expressions \(n\in \widehat{a_{\delta}}\) or \(n\notin \widehat{a_{\delta}}\). A condition \(P\) forces \(n\in \widehat{a_{\delta}}\) if \((n\in a_{\delta})\in P\). Similarly, we can define the relation “\(P\) forces \(\varphi\)” for any condition \(P\) and any formula \(\varphi\) of the forcing language. This definition is carried out inside \(M\) and has the following properties

(a) for each \(\varphi\), no \(P\) forces both \(\varphi\) and \(\neg \varphi\);

(b) if \(P\) forces \(\varphi\) and \(Q\supset P\) then \(Q\) forces \(\varphi\);

(c) for each \(\varphi\) and each \(P\), there is \(Q\supset P\) which decides \(\varphi\) (i.e. \(Q\) forces \(\varphi\) or \(Q\) forces \(\neg \varphi\)). Since \(M\) is countable, there is a sequence \(P_0\subseteq P_1\subseteq \dots P_s\dots \) (outside \(M\)) such that each formula is decided by some \(P_s\). The extension \(N\) is then obtained by adjoining to \(M\) the sequence \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) where \(a_{\delta}=\{n:n\in a_{\delta}\text{ belongs to some } P_s\}\). The significance of the forcing method in the construction of \(N\) is expressed by the following

Lemma: A formula is true in \(N\) if and only if it is forced by some \(P_s\). Using this, the author proves that \(N\) is a model of ZF + Axiom of Choice and that \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) is a sequence of distinct sets of integers. The proof is completed when verified that every cardinal number in \(M\) is also a cardinal number in \(N\), so that \(N\) satisfies \(2^{\aleph_0}>\aleph_1\).

Finally, it is shown how the construction described above yields the relative consistency proof of ZF + Axiom of Choice + \(2^{\aleph_0}>\aleph_1\). To verify the truth of a statement in \(N\) we need the truth of only finitely many axioms in \(M\); and since every finite collection of axioms of ZF has a countable transitive model, every contradiction in ZF + AC + \(2^{\aleph_0}>\aleph_1\) leads to a contradiction in ZF.

The author starts with an assumption that there is a countable transitive model \(M\) of Zermelo-Fraenkel set theory ZF which satisfies the axiom of constructibility and constructs an extension \(N\) of \(M\) which is a model of ZF + Axiom of Choice+\(2^{\aleph_0}>\aleph_1\). The idea is to obtain \(N\) by adjoining to \(M\) a sequence \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) of “generic” sets of integers, where \(\aleph_{\tau}\) is a cardinal number in \(M\), greater than \(\aleph_1\).

The key device is the notion of forcing. The forcing language consists of names of all elements of \(M\) and of generic sets \(a_\delta\), and of expressions using logical symbols and set-theoretical operations. A condition is a finite consistent set of expressions \(n\in \widehat{a_{\delta}}\) or \(n\notin \widehat{a_{\delta}}\). A condition \(P\) forces \(n\in \widehat{a_{\delta}}\) if \((n\in a_{\delta})\in P\). Similarly, we can define the relation “\(P\) forces \(\varphi\)” for any condition \(P\) and any formula \(\varphi\) of the forcing language. This definition is carried out inside \(M\) and has the following properties

(a) for each \(\varphi\), no \(P\) forces both \(\varphi\) and \(\neg \varphi\);

(b) if \(P\) forces \(\varphi\) and \(Q\supset P\) then \(Q\) forces \(\varphi\);

(c) for each \(\varphi\) and each \(P\), there is \(Q\supset P\) which decides \(\varphi\) (i.e. \(Q\) forces \(\varphi\) or \(Q\) forces \(\neg \varphi\)). Since \(M\) is countable, there is a sequence \(P_0\subseteq P_1\subseteq \dots P_s\dots \) (outside \(M\)) such that each formula is decided by some \(P_s\). The extension \(N\) is then obtained by adjoining to \(M\) the sequence \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) where \(a_{\delta}=\{n:n\in a_{\delta}\text{ belongs to some } P_s\}\). The significance of the forcing method in the construction of \(N\) is expressed by the following

Lemma: A formula is true in \(N\) if and only if it is forced by some \(P_s\). Using this, the author proves that \(N\) is a model of ZF + Axiom of Choice and that \(\{a_{\delta}: \delta<\aleph_{\tau}\}\) is a sequence of distinct sets of integers. The proof is completed when verified that every cardinal number in \(M\) is also a cardinal number in \(N\), so that \(N\) satisfies \(2^{\aleph_0}>\aleph_1\).

Finally, it is shown how the construction described above yields the relative consistency proof of ZF + Axiom of Choice + \(2^{\aleph_0}>\aleph_1\). To verify the truth of a statement in \(N\) we need the truth of only finitely many axioms in \(M\); and since every finite collection of axioms of ZF has a countable transitive model, every contradiction in ZF + AC + \(2^{\aleph_0}>\aleph_1\) leads to a contradiction in ZF.

Reviewer: Tomas Jech

##### MSC:

03Exx | Set theory |