Forcing and Independence — Mathematics Knowledge System

Overview

Forcing is Paul Cohen's revolutionary technique (1963) for constructing models of set theory in which specific statements — notably the Continuum Hypothesis and the Axiom of Choice — have chosen truth values. It established that the Continuum Hypothesis is independent of ZFC, completing Gödel's earlier consistency result.

Intuition

Forcing adjoins a 'generic' object $G$ to a ground model $M$ — like adjoining a transcendental to a field — creating an extension $M[G]$. The forcing poset $\mathbb{P}$ controls what $G$ looks like. By choosing $\mathbb{P}$ carefully (e.g., adding $\aleph_2$ many subsets of $\omega$), one can make $2^{\aleph_0} = \aleph_2$ in $M[G]$, violating CH.

Formal Definition

Definition

p \Vdash \varphi \quad (\text{``}p \text{ forces } \varphi\text{''})

The forcing relation: condition p forces sentence φ to hold in all generic extensions

forcing-relation

M[G] = \{ \tau_G : \tau \in M^\mathbb{P} \}

Generic extension: domain is set of evaluations of P-names in M under the generic filter G

generic-extension

\text{Con}(\text{ZFC}) \implies \text{Con}(\text{ZFC} + \neg\text{CH})

Cohen's main result: the negation of CH is consistent with ZFC

independence-ch

\text{Add}(\omega, \aleph_2) = \{ p : \aleph_2 \times \omega \to 2 : |p| < \omega \}

Cohen's forcing poset for adding aleph_2 many new subsets of omega

cohen-forcing

Notation

Notation	Meaning
$p \Vdash \varphi$	Condition p forces formula φ
$M[G]$	Generic extension of ground model M by generic filter G
$\mathbb{P}$	Forcing poset (partially ordered set with maximum element)
$G$	P-generic filter over M: an M-generic ultrafilter on P
$\tau_G$	Interpretation (evaluation) of a P-name τ in M[G]

Theorems

Theorem 1: The Forcing Theorem

If G is \mathbb{P}-generic over M and \varphi is a sentence of set theory, then M[G] \models \varphi iff \exists p \in G,\, p \Vdash \varphi. Moreover, the forcing relation \Vdash is definable in M.

Theorem 2: Independence of the Continuum Hypothesis

Both CH and \neg CH are consistent with ZFC (assuming ZFC is consistent). Gödel proved Con(ZFC + CH) via \mathbf{L}; Cohen proved Con(ZFC + \neg CH) via forcing.

Theorem 3: Preservation of Cardinals

If \mathbb{P} satisfies the countable chain condition (ccc) — every antichain is countable — then M[G] has the same cardinals as M.

Worked Examples

Take $\mathbb{P} = \text{Add}(\omega, \aleph_2)$: finite partial functions from $\aleph_2 \times \omega$ to $\{0,1\}$. A generic filter $G$ yields a total function $f_G : \aleph_2 \times \omega \to \{0,1\}$.
For each $\alpha < \aleph_2$, define $r_\alpha = \{n : f_G(\alpha, n) = 1\} \subseteq \omega$. These $\aleph_2$ many reals $r_\alpha$ are all distinct (a density argument shows $r_\alpha \neq r_\beta$ for $\alpha \neq \beta$).
$\{r_\alpha : \alpha < \aleph_2\} \subseteq \mathcal{P}(\omega)$
Since $\mathbb{P}$ is ccc, $\aleph_1^{M[G]} = \aleph_1^M$ and $\aleph_2^{M[G]} = \aleph_2^M$. In $M[G]$, there are $\aleph_2$ many distinct subsets of $\omega$, so $2^{\aleph_0} \geq \aleph_2 > \aleph_1$, negating CH.

Answer: Cohen forcing adds $\aleph_2$ many new reals without collapsing cardinals, making $2^{\aleph_0} \geq \aleph_2$ and refuting CH.

Practice Problems

Difficulty 8/10

State what it means for $G$ to be $\mathbb{P}$-generic over $M$, and why genericity is needed.

Difficulty 9/10

Explain the countable chain condition (ccc) and why it implies cardinal preservation.

Difficulty 10/10

Describe Gödel's constructible universe $\mathbf{L}$ and how it establishes Con(ZFC + CH).

Historical Background

Cantor's Continuum Hypothesis (1878) asked whether $2^{\aleph_0} = \aleph_1$. Gödel showed CH is consistent with ZFC in 1938 by constructing the constructible universe $L$. Cohen invented forcing in 1963 to show $\neg\text{CH}$ is also consistent — making CH independent. Cohen was awarded the Fields Medal in 1966. Forcing has since become the primary method for proving independence results in set theory.

1878
Cantor conjectures the Continuum Hypothesis
Georg Cantor
1938
Gödel proves Con(ZFC) → Con(ZFC + CH) via the constructible universe L
Kurt Gödel
1963
Cohen invents forcing and proves Con(ZFC) → Con(ZFC + ¬CH)
Paul Cohen
1966
Cohen awarded Fields Medal for the independence of CH
Paul Cohen

Summary

Forcing is a method to build new models of set theory (generic extensions $M[G]$) from a ground model $M$.
The forcing poset $\mathbb{P}$ controls what new sets are added; conditions in $\mathbb{P}$ 'force' sentences.
Cohen used forcing to show $\neg\text{CH}$ is consistent with ZFC, completing the independence of CH.
The ccc condition on $\mathbb{P}$ ensures cardinals are not collapsed in the extension.
Forcing is now the standard tool for proving independence results in set theory.

References

BookKunen, K. Set Theory: An Introduction to Independence Proofs. Elsevier, 1980.
BookJech, T. Set Theory (3rd ed.). Springer, 2003.