Forcing

Origin. Cohen (1963). Proved the independence of the Continuum Hypothesis from ZFC and of the Axiom of Choice from ZF. The two base theories are different and must be kept apart: AC is one of ZFC's own axioms, so it cannot be independent of ZFC; the choiceless theory ZF is what Cohen's symmetric-submodel argument works over. CH is independent of ZFC proper. A general method for building models of set theory with prescribed properties.

Models. Extend a countable transitive model of ZFC by adjoining a generic filter for a partial order of "conditions." The extension satisfies ZFC and can be made to satisfy or refute a target statement.

Formalism.

Forcing poset: (P, ≤): conditions, with q ≤ p meaning q is stronger (more information).

Generic filter: G ⊆ P a filter meeting every dense set that lies in the ground model M. Exists for countable M (not in M).

Names and interpretation: P-names τ built recursively. val_G(τ) interprets names in the extension M[G].

Forcing relation: p ⊩ φ, definable in M. Truth lemma: M[G] ⊨ φ iff some p ∈ G forces φ.

Cohen forcing: Finite partial functions adding new reals ⟹ M[G] ⊨ ¬CH. Collapsing and Lévy forcing adjust cardinals.

Preservation: ccc (countable chain condition) posets preserve cardinals and cofinalities. Iterated forcing with support builds complex models.

Boolean-valued models: V^B over a complete Boolean algebra B. Truth values in B; generic = ultrafilter.

Symbols.

Symbol	Unicode	Name	Meaning
⊩	U+22A9	Forces	Forcing relation
P, ≤	—	Poset	Conditions
G	—	Generic filter	Meets all dense sets
M[G]	—	Extension	Generic extension
V^B	—	Boolean model	Boolean-valued universe

Metatheory. M[G] satisfies ZFC (forcing preserves the axioms). Con(ZFC) ⟹ Con(ZFC + ¬CH); with the constructible universe giving Con(ZFC + CH), CH is independent. ccc guarantees cardinal preservation; proper forcing generalizes it.

Applies to. Independence and relative-consistency proofs. Cardinal arithmetic. Set-theoretic topology and combinatorics. Large-cardinal and determinacy consistency results.

Limitations. Requires a ground model; yields relative consistency, not truth. Technically intricate (names, genericity, iteration). Does not decide which axioms are "correct."