Forcing

Forcing, introduced by Paul Cohen in 1963, is the primary technique for proving independence results in set theory. It constructs new models of set theory (generic extensions) by adjoining "generic" objects, analogous to field extensions in algebra.

Forcing Posets

Definition7.3Forcing Poset

A forcing notion (or partial order) is a triple $(\mathbb{P}, \leq, \mathbf{1})$ where $\mathbb{P}$ is a set of conditions, $\leq$ is a partial order on $\mathbb{P}$ ( $p \leq q$ means $p$ extends/is stronger than $q$ ), and $\mathbf{1}$ is the weakest condition. Two conditions $p, q$ are compatible if there exists $r$ with $r \leq p$ and $r \leq q$ .

Definition7.4Generic Filter

A set $D \subseteq \mathbb{P}$ is dense if for every $p \in \mathbb{P}$ , there exists $q \in D$ with $q \leq p$ . A filter $G \subseteq \mathbb{P}$ is $M$ -generic (for a ground model $M$ ) if $G$ meets every dense set $D \in M$ . The generic extension $M[G]$ is the smallest model of ZFC containing both $M$ and $G$ .

Cohen Forcing

ExampleAdding a Cohen Real

Cohen forcing uses $\mathbb{P} = \mathrm{Fin}(\omega, 2)$ : conditions are finite partial functions from $\omega$ to $\{0, 1\}$ , ordered by reverse inclusion. A generic filter $G$ determines a new function $g: \omega \to \{0,1\}$ (a "Cohen real") not in the ground model. This shows that the continuum hypothesis can fail: by adding $\aleph_2$ many Cohen reals to a model of CH, one obtains a model where $2^{\aleph_0} = \aleph_2$ .

RemarkThe Forcing Relation

The forcing relation $p \Vdash \varphi$ means "condition $p$ forces that $\varphi$ holds in the generic extension." This is defined recursively on formulas and satisfies: if $p \Vdash \varphi$ and $G$ is generic with $p \in G$ , then $M[G] \models \varphi$ . The truth lemma states: $M[G] \models \varphi$ iff some $p \in G$ forces $\varphi$ .

Definition7.5Chain Condition

A forcing $\mathbb{P}$ satisfies the countable chain condition (c.c.c.) if every antichain in $\mathbb{P}$ (set of pairwise incompatible conditions) is countable. c.c.c. forcing preserves all cardinals from the ground model, which is crucial for controlling the cardinal structure of the extension.