Cohen Forcing in Set Theory

• Cohen forcing is a set theoretic method that uses posets of finite partial functions to add generic subsets and create models where axioms like CH and AC vary.
• It employs chain conditions such as c.c.c. to preserve cardinalities and cofinalities while carefully managing generic extensions and combinatorial properties.
• Its adaptations to large cardinals, modal logic, and choiceless models highlight its versatility in proving independence results and analyzing structural set theory.

Cohen forcing is a central method in set theory for constructing models in which the continuum hypothesis, the axiom of choice, or other combinatorial and cardinal characteristics fail or change. The defining feature of Cohen forcing is its simplicity and power: it adds generic subsets, typically of countable sets, while controlling cardinal arithmetic, cofinalities, and absoluteness properties in the resulting generic extensions. The formalism has been adapted and elaborated for large cardinals, choiceless models, and combinatorial investigations—serving as a paradigmatic forcing in both foundational research and modal logic analyses.

1. Fundamental Definitions and Constructions

Cohen forcing, usually denoted $\mathbb{P}_C$ or $\mathrm{Add}(\kappa,\lambda)$, is defined as the poset of partial functions $p\!: S \to 2$ from some set $S$ (often $\omega$ or $\kappa\times\lambda$) to $\{0,1\}$, with $|\dom p|$ finite or $|\dom p|<\kappa$ for uncountable supports, ordered by extension: $q \leq p$ iff $q \supseteq p$. For example:

• For adding a single Cohen real: $\mathbb{P}_C = \{ p: \dom(p) \subseteq \omega, \ |\dom(p)| < \omega, \ p: \dom(p) \to \{0,1\} \}$.
• For adding $\kappa^+$-many Cohen subsets to $\kappa$: $\mathrm{Add}(\kappa,\kappa^+)$ consists of all $p: \kappa \times \kappa^+ \to \{0,1\}$ with $|\dom(p)| < \kappa$, ordered by extension; in a generic extension, this adds $\kappa^+$-many distinct generic subsets of $\kappa$ (Benhamou et al., 2022).

The generic extension $V[G]$ is defined via evaluation of $\mathbb{P}$-names with the generic filter $G$, and includes the sets coded by $G$. The existence and properties of generic filters in this context are established via the Rasiowa–Sikorski lemma, Dependent Choice, and recursive constructions of the value and name maps (Gunther et al., 2018).

2. Forcing Properties: Chain Conditions and Preservation

Cohen forcing exemplifies important preservation properties:

• In ZFC, Cohen forcing is c.c.c. (every antichain is countable) and preserves cardinals and cofinalities above $\omega_1$.
• In ZF (without Choice), standard c.c.c. is insufficient: finite-support products of c.c.c. posets can collapse $\omega_1$. Instead, the “narrowness” property is defined. A poset $P$ is $(\theta,\nu)$-narrow if for any $\mu\leq\nu$, any sequence of partial $\parallel$-homomorphisms $f_i: P \rightharpoonup \mathrm{Ord}$, the union of the ranges has cardinality $\leq\max\{\theta,\mu\}$ (Ikegami et al., 2022).

Finite-support products and iterations of Cohen forcing are uniformly $\omega$-narrow, so they preserve $\omega_1$ and higher cardinals in arbitrary models of ZF (Ikegami et al., 2022).

3. Specializations: Dedekind-Finite Sets and Choiceless Models

Cohen forcing’s effect on cardinal characteristics becomes nuanced over models lacking the axiom of choice. Notably:

• In Cohen’s first model (a symmetric submodel of $V[G]$), there exists a Dedekind-finite infinite set of reals $A$ (Karagila et al., 2019).
• Forcing with $\mathrm{Col}_{\mathrm{inj}}(\kappa,A)$ (adding an injection $f:\kappa \to A$) is forcing-equivalent to adding $\kappa$ Cohen reals; the resulting model collapses the Dedekind-finiteness.
• Forcing with $\mathrm{Col}(A,\kappa)$ (adding a surjection $f:A \to \kappa$) adds no new reals and preserves the Dedekind-finiteness of $A$.

Karagila–Schlicht (Karagila et al., 2019) provide a combinatorial characterization for when forcing with $\mathrm{Add}(A,1)$ (adding a Cohen subset of $A$) preserves Dedekind-finiteness:

• $[A]^{<\omega}$ is Dedekind-finite.
• $\mathrm{Add}(A,1)$ has no infinite antichains or countable infinite antichains.
• $\mathrm{Add}(A,1)$ does not add a new real or subset of ordinals.
• $2^A$ is extremally disconnected.

These conditions, and their equivalence, reveal deep connections between forcing, combinatorics, and topological properties in choiceless contexts.

4. Modal Logic and Structural Principles

Cohen forcing has been used to analyze the modal logic of forcing: the system of valid schematic principles governing truth in all (possibly iterated) Cohen-generic extensions. The class of all Cohen posets (finite-support products of “add Cohen subsets to some $X$”) yields exactly the modal logic S4.3 (Hamkins et al., 2012):

• Modal operators are interpreted as: $\Box\varphi$ is “$\varphi$ holds in all Cohen-generic extensions”; $\Diamond\varphi$ is “$\varphi$ holds in some Cohen-generic extension”.
• S4.3 includes: K ($\Box(\varphi\to\psi)\to(\Box\varphi\to\Box\psi)$), T ($\Box\varphi\to\varphi$), 4 ($\Box\varphi\to\Box\Box\varphi$), and .3 ($\Diamond\Box\varphi\to\Box\Diamond\varphi$).

The completeness is witnessed via “ratchets”: families of statements $r_\alpha$ (for $\alpha>0$) such that $r_{\alpha+1} \to r_\alpha$, and any Cohen extension can push the ratchet to any prescribed ordinal value; e.g., $r_\alpha: 2^\omega > \aleph^L_\alpha$ in $L$ (Hamkins et al., 2012).

5. Large Cardinals and Forcing Interactions

Cohen forcing can be systematically integrated into sophisticated large cardinal and Prikry-type forcings. A notable result (Benhamou et al., 2022):

• With a single measurable cardinal $\kappa$, a Prikry-type Easton support iteration is constructed such that in the resulting model, $\kappa^+$-many Cohen subsets are added to $\kappa$.
• The iteration involves forcing with lottery sums of Cohen forcings at inaccessibles below $\kappa$, and $\mathrm{Add}(\kappa,\kappa^+)$ at $\kappa$.
• There exists a $\kappa$-complete ultrafilter $W$ in the extension, so that $\mathrm{Pri}(W)$ forces the existence of a $V^*$-generic for $\mathrm{Add}(\kappa,\kappa^+)$.
• The construction provides strong non-Galvin witness families and improves the large cardinal bounds for certain combinatorial phenomena from supercompact to measurable.

Extender-based Prikry forcings, including the Merimovich and Gitik–Magidor variants, demonstrate that whenever $\kappa^+$-many subsets of $\kappa$ are added, $\kappa$ becomes singular of cofinality $\omega$ in the intermediate model; in particular, no intermediate model can both restore the measurability of $\kappa$ and raise $2^\kappa\geq\kappa^{++}$ (Benhamou et al., 2022).

6. Technical Formalization and Implementation

The syntax and semantics of Cohen forcing are suitable for formalization in systems such as Isabelle/ZF (Gunther et al., 2018), yielding a rigorous encoding of:

• Forcing notions as preorders with a top element.
• Dense subsets and M-generic filters.
• Rasiowa–Sikorski lemma for dense sets in countable settings.
• Recursive construction of the generic extension $M[G]$, canonical names, and extension of foundational axioms (pairing, transitivity) from $M$ to $M[G]$.

The framework supports the “names/evaluation” machinery necessary for tracking extensions and verifying structural preservation of set-theoretic axioms.

7. Impact and Broader Consequences

Cohen forcing is fundamental for the analysis and manipulation of the continuum, the independence of CH and the axiom of choice, and the construction of pathological or “unexpected” models—for instance, models with Dedekind-finite sets of reals, or generic extensions in which key combinatorial or large cardinal properties are controlled. It is also essential in generic absoluteness debates: even highly restricted classes (e.g., all finite-support products of Cohen forcing) can exhibit profound non-absoluteness, such as collapsing cardinals’ cofinalities under certain absoluteness principles (Ikegami et al., 2022). The modal, combinatorial, and model-theoretic properties shaped by Cohen forcing continue to illuminate the structure and limitations of the set-theoretic universe.