Forcing method

A special method for constructing models of axiomatic set theory. It was proposed by P.J. Cohen in 1963 to prove the compatibility of the negation of the continuum hypothesis, $\neg \mathsf{CH}$, and other set-theoretic assumptions with the axioms of the Zermelo–Fraenkel system $\mathsf{ZF}$ ([1]). The forcing method was subsequently simplified and modernized ([2]–[6]). In particular, it was found that the method was related to the theory of Boolean-valued models ([2], [3]) and Kripke models ([6]).

The central concept of the forcing method is the forcing relation $$p \Vdash \phi.$$ (This is to be read as “the condition $p$ forces the formula $\phi$”.)

The definition of the forcing relation is preceded by the specification of a language $\mathcal{L}$ and a partially ordered set $\mathbb{P}$ of forcing conditions $p$ with an order relation $\leq$. The language $\mathcal{L}$ may contain variables and constants of different sorts (or types).

The construction of the model of $\mathsf{ZF}$ proposed by Cohen in which $\mathsf{CH}$ is violated proceeds as follows. A set $M$ is called transitive if $$x \in M \Rightarrow x \subseteq M.$$

Let $M$ be a denumerable transitive set that is a model of $\mathsf{ZF}$, and let $\lambda \in M$ be an ordinal number in the sense of von Neumann, i.e., $\lambda = \{ \alpha \mid \alpha < \lambda \}$. Let $A \subseteq \lambda \times \omega_{0}$ be an arbitrary set (possibly $A \notin M$), where $\omega_{0}$ denotes the first infinite ordinal number. If $X$ is a transitive set, then let $\operatorname{Def}(X)$ denote the set of all $X$-definable subsets, i.e., $\operatorname{Def}(X) = \operatorname{Def}(X,\in \!\! |_{X \times X})$. A process similar to that of defining Gödel constructive sets is used to define, via transfinite induction, a set ${M_{\alpha}}[A]$ for any ordinal number $\alpha$: $${M_{\alpha}}[A] \stackrel{\text{df}}{=} \bigcup_{\beta < \alpha} \operatorname{Def}({M_{\beta}}[A]) \cup \{ x \in M \cup \{ A\} \mid x \subseteq {M_{\beta}}[A] \}.$$

Let $M[A] \stackrel{\text{df}}{=} {M_{\alpha_{0}}}[A]$, where $a_{0} \stackrel{\text{df}}{=} \{ \alpha \in M \mid \alpha \text{ is an ordinal number} \}$. The model of $\mathsf{ZF}$ in which $\mathsf{CH}$ is violated is to be found among models of the type $M[A]$. Let $\lambda$ be an ordinal number such that the statement “$\lambda$ is the second uncountable ordinal” is true in $M$.

The set of forcing conditions $\mathbb{P}$ and the relation $\leq$ are defined by the following equivalences: (a) $p \in \mathbb{P} \iff p$ is a function defined on some finite subset of $\lambda \times \omega_{0}$, with values in $\{ 0,1 \}$; (b) $p \leq q \iff q$ is an extension of $p$. The language $\mathcal{L}$ considered is a so-called ramified language, with many types of variables (each $\alpha \leq \alpha_{0}$ having its own type of variables running through the set ${M_{\alpha}}[A]$) and with names (i.e., individual constants) for each set from $M[A]$. If $x \in M$, then the name of $x$ is denoted by $x^{\vee}$. Let $a$ be the name of the set $A$. The forcing relation $p \Vdash \phi$ is introduced by a definition involving transfinite induction, which has, in particular, the following characteristics:

1. $(p \Vdash \langle \delta,n \rangle^{\vee} \in a) \iff (p(\langle \delta,n \rangle) = \mathbf{1})$;
2. $(p \Vdash \neg \phi) \iff \neg (\exists q \geq p)(q \Vdash \phi)$;
3. $(p \Vdash (\phi \lor \psi)) \iff ((p \Vdash \phi) \lor (p \Vdash \psi))$;
4. $(p \Vdash (\phi \land \psi)) \iff ((p \Vdash \phi) \land (p \Vdash \psi))$;
5. If $\alpha$ is the type of the variable $x$, then $(p \Vdash (\exists x) \phi(x)) \iff (\exists c \in C_{\alpha})(p \Vdash \phi(c))$, where $C_{\alpha}$ is the set of all constants of type $\alpha$.

A sequence of forcing conditions in $\mathbb{P}$, $$p_{0} \leq \ldots \leq p_{n} \leq \ldots,$$ is called complete if for any closed formula $\phi$ of $\mathcal{L}$, one has $$\exists n \in \omega_{0}: \qquad (p_{n} \Vdash \phi) \lor (p_{n} \Vdash \neg \phi).$$ The countability of the set of all closed formulas of $\mathcal{L}$, together with (2), makes it possible to demonstrate the existence of a complete sequence that begins with an arbitrary forcing condition $p_{0} \in \mathbb{P}$.

A set $A$ contained in $\lambda \times \omega_{0}$ is called generic with respect to the model $M$ if there exists a complete sequence $(p_{n})_{n \in \omega_{0}}$ such that $\displaystyle \bigcup_{n < \omega_{0}} p_{n}$ is the characteristic function $\chi_{A}$ of $A$. The following two facts about generic sets and the forcing relation are of fundamental importance:

• If $A$ is a generic set, then $$M[A] \models \phi \iff (\exists p \subseteq \chi_{A})(p \Vdash \phi),$$ where $M[A] \models \phi$ means that formula $\phi$ is true in $M[A]$.
• The relation $$p \Vdash \phi(c_{1},\ldots,c_{n}), \qquad (\text{where } c_{1},\ldots,c_{n} \text{ are constants in } \mathcal{L})$$ regarded as a relation among $p,c_{1},\ldots,c_{n}$, can be expressed in the model $M$.

In view of the aforementioned facts, it is sufficient to show that the statement $$(\forall p)(p \Vdash \neg \neg \phi), \qquad \text{i.e.}, \qquad (\forall p)(\exists q \geq p)(q \Vdash \phi),$$ is true in the model $M$, i.e., to prove that $M[A] \models \phi$. The verification of the validity of the axioms of $\mathsf{ZF}$ and $\neg \mathsf{CH}$ in the model $M[A]$ is based upon this proposition. The verification of $\neg \mathsf{CH}$ in $M[A]$ also involves the use of special properties of the set of forcing conditions, which makes it possible to show that:

1. If the ordinal numbers $\delta_{1},\delta_{2} < \lambda$ are unequal, then one has $$(\forall p)(\exists q \geq p)(\exists n < \omega_{0})(q(\langle \delta_{1},n \rangle) \neq q(\langle \delta_{2},n \rangle)),$$ that is, $$A_{\delta_{1}} \neq A_{\delta_{2}}, \qquad \text{where } A_{\delta} \stackrel{\text{df}}{=} \{ n < \omega_{0} \mid \langle \delta,n \rangle \in A \}.$$
2. $M[A] \models (\lambda \text{ is the second uncountable ordinal})$.

There is the following relation between forcing and Boolean-valued models: If one introduces the notation $$\begin{gather} \| \phi \| \stackrel{\text{df}}{=} \{ p \in \mathbb{P} \mid p \Vdash \neg \neg \phi \}, \\ \mathbb{B} \stackrel{\text{df}}{=} \{ X \subseteq \mathbb{P} \mid (\forall p)((p \in X) \iff (\forall q \geq p)(\exists r \geq q)(r \in X)) \}, \end{gather}$$ then $(\mathbb{B},\subseteq)$ is a complete Boolean algebra, and $\| \phi \| \in \mathbb{B}$ is the Boolean value of the formula $\phi$. Hence, to specify a partially ordered set $(\mathbb{P},\leq)$ and to define the relation $p \Vdash \neg \neg \phi$ is equivalent to constructing some Boolean-valued model $\mathfrak{M}$. The analysis of the proof of theorems of the type $$M[A] \models \phi_{0},$$ where $\phi_{0}$ is an axiom of $\mathsf{ZF}$ or $\neg \mathsf{CH}$, leads to the conclusion that the formulas of $\mathsf{ZF}$ expressing the theorem $$(\forall p)(p \Vdash \neg \neg \phi_{0}),$$ i.e., $\| \phi_{0} \| = 1_{\mathbb{B}}$, are deducible from the axioms of $\mathsf{ZF}$. Therefore, $\mathfrak{M}$ is a Boolean-valued model for $\mathsf{ZF} + \neg \mathsf{CH}$, constructed solely with the tools of $\mathsf{ZF}$. The assumption of countability of a transitive set that is a model of $\mathsf{ZF}$ and the concept of a generic set are both inessential in the relative consistency proof.

It has been shown that the construction of the Boolean-valued model may be simplified ([2], [3], [5]). In particular, the introduction of the ramified language $\mathcal{L}$ is not necessary. A generic model $M[A]$ may also be constructed as follows ([4]):

A subset $X$ of a partially ordered set $(\mathbb{P},\leq)$ is called dense if $$(\forall p)(\exists q \geq p)(q \in X).$$ Let $\mathbb{P}$ and the relation $\leq$ be elements of some denumerable transitive set $M$ that is a model of $\mathsf{ZF}$. A subset $G \subseteq \mathbb{P}$ is called an $M$-generic filter if

1. $((p \in G) \land (q \leq p)) \Rightarrow (q \in G)$;
2. $((p \in G) \land (q \in G)) \Rightarrow (\exists r \in G)((p \leq r) \land (q \leq r))$;
3. $((X \in M) \land (X \text{ is dense in } \mathbb{P})) \Rightarrow X \cap G \neq \varnothing$.

Let $G$ be an $M$-generic filter in $\mathbb{P}$. As $M$ is denumerable, $G$ exists. In general, $G \notin M$. A relation $\in_{G}$ is then defined by the equivalence $$x \in_{G} y \iff (\exists p \in G)(\langle x,p \rangle \in y),$$ where $x$ and $y$ are arbitrary elements of the model $M$.

Let a function $F_{G}$ be defined on $M$ by the equations $${F_{G}}(y) \stackrel{\text{df}}{=} \{ {F_{G}}(x) \mid x \in_{G} y \},$$ and let $$N_{G} \stackrel{\text{df}}{=} \{ {F_{G}}(x) \mid x \in M \}.$$ If $\phi$ is a closed formula of the language $\mathsf{ZF}$, supplemented by constants to denote each set from $M$, then one defines $$p \Vdash \phi \iff (\forall G \ni p)(G \text{ is an } M\text{-generic filter} \Rightarrow N_{G} \models \phi).$$ It can be now shown that:

(I) $N_{G} \models \phi \iff (\exists p \in G)(p \Vdash \phi)$.

(II) The relation $p \Vdash \phi(c_{1},\ldots,c_{n})$ can be defined in the model $M$ for each formula $\phi$.

By using solely (I) and (II), and basing oneself on the fact that $M$ is a model of $\mathsf{ZF}$, it is possible to establish that $N_{G}$ is a model of $\mathsf{ZF}$. If $\mathbb{P}$ is defined by the equivalences (a) and (b) given much earlier on, then one has $N_{G} \models \neg \mathsf{CH}$, that $\displaystyle \bigcup G$ is the characteristic function of some set $A \subseteq \lambda \times \omega_{0}$, and $M[A] = N_{G}$. The relation $p \Vdash \phi$, which is definable in $M$, does not satisfy points (3) and (5) of Cohen’s definition of the forcing relation. One has instead $$(p \Vdash (\exists x) \phi(x)) \iff (\forall p' \geq p)(\exists p'' \geq p')(\exists a \in M)(p'' \Vdash \phi(a)).$$ If it is assumed that $$\| \phi \| \stackrel{\text{df}}{=} \{ p \in \mathbb{P} \mid p \Vdash \phi \},$$ then a Boolean-valued model for $\mathsf{ZF} + \neg \mathsf{CH}$ is obtained that is definable in $M$ and has a Boolean algebra $\mathbb{B}$ that is identical with that of Cohen.

Therefore, the forcing method consists, in fact, of constructing a $\mathbb{B}$-valued model and a homomorphism, which preserves certain infinite unions and intersections of the algebra $\mathbb{B}$, into the two-element algebra $\{ 0,1 \}$. For applications of the forcing method in set theory see, for example, [2].

References

[1]	P.J. Cohen, “Set theory and the continuum hypothesis”, Benjamin (1966).
[2]	T.J. Jech, “Lectures in set theory: with particular emphasis on the method of forcing”, Lect. notes in math., 217, Springer (1971).
[3]	G. Takeuti, W.M. Zaring, “Axiomatic set theory”, Springer (1973).
[4]	J.R. Shoenfield, “Mathematical logic”, Addison-Wesley (1967).
[5]	Yu.I. Manin, “The problem of the continuum”, J. Soviet Math., 5: 4 (1976), pp. 451–502; Itogi Nauk. i Tekhn. Sovrem. Problemy, 5 (1975), pp. 5–73.
[6]	M.C. Fitting, “Intuitionistic logic, model theory and forcing”, North-Holland (1969).

Comments

For a modern elementary introduction with applications, see [a1]. A quicker, though incomplete, account is [a3]. [a2] presents the Boolean-valued approach mentioned above.

References

[a1]	K. Kunen, “Set theory, an introduction to independence proofs”, North-Holland (1980).
[a2]	J.L. Bell, “Boolean-valued models and independence proofs in set theory”, Oxford Univ. Press (1977).
[a3]	J.P. Burgess, “Forcing”, J. Barwise (ed.), Handbook of mathematical logic, North-Holland (1977), pp. 403–452.
[a4]	T.J. Jech, “Set theory”, Acad. Press (1978), pp. 523ff. (Translated from German)