Forcing I: Posets, Generic Filters, Names, and the Fundamental Theorem

Intuition Beginner

The previous unit shrank the universe of sets: it kept only the sets you could describe, building the slim world $L$ where the continuum hypothesis is true. Forcing does the opposite. You start with a fixed world of sets and you grow it by carefully adding one new object that was not there before — and you arrange the new object so that the enlarged world still obeys all the rules of set theory, yet now answers some open question the old world left open.

The new object is built from tiny promises called conditions. Each condition is a finite piece of information about the object you are adding — say, the first few digits of a brand-new infinite sequence. A stronger condition is one that promises more. To actually pin down the new object you make an infinite sequence of ever-stronger promises that is generic: it dodges no requirement. For every meaningful demand the old world can phrase, your sequence eventually settles it. Think of assembling a jigsaw where the only rule is that you must keep placing pieces until every region anyone could point to is covered.

The surprise is that this generic object cannot already live in the old world — it is genuinely new — yet the old world can still talk about it in advance, naming the thing it is building before it exists. That dual fact, new but nameable, is the engine that lets forcing add a set witnessing the failure of a hypothesis while keeping every axiom intact.

Visual Beginner

A condition is a finite promise; stronger conditions sit lower because they promise more. A generic sequence walks downward through the tree of promises, and the one rule is that it must enter every region the old world can mark out (every dense set).

            1   (empty promise, decides nothing)
           / \
        p0     p1            level 1: one value decided
        / \    / \
     q00 q01 q10 q11         level 2: two values decided
      .   .   .   .
   stronger conditions are LOWER (they promise more)

   a DENSE region: a set of conditions you can always reach by
   going further down, no matter where you start

   a GENERIC path: a downward route that enters EVERY dense region
   the old world can name -> it pins down one brand-new object

The table contrasts the old world with the grown one. The new object and any sets the rules then demand are the only genuinely new arrivals; everything old is carried across unchanged under a copy name.

feature	old world (ground model)	grown world (generic extension)
every old set	present	present, under a copy name
the new generic object	absent	present
the axioms of set theory	all hold	all still hold
the open question	undecided	now decided by the new object

The key idea: a generic descending path through the tree of finite promises adds one new set, and naming-in-advance lets the old world prove the grown world still obeys every axiom.

Worked example Beginner

Let us add one brand-new infinite sequence of $0$s and $1$s — a Cohen real — and watch the promises do the work. A condition is a finite list that fixes the sequence at a few spots and says nothing elsewhere.

Step 1. Start with the empty promise: it fixes no value. Write it as the blank list. It is the weakest condition, sitting at the top.

Step 2. A stronger condition fixes more. For instance "spot $0$ is $1$, spot $1$ is $0$" is stronger than "spot $0$ is $1$" because it keeps that promise and adds another. Lengthening a list, never contradicting it, is what "stronger" means here.

Step 3. Consider a demand the old world can phrase: "the sequence has a $1$ somewhere past spot $5$." The conditions that settle this demand form a region you can always reach — from any finite list, just extend it by putting a $1$ at, say, spot $6$. So this region is dense: reachable from anywhere by going further down.

Step 4. A generic descending sequence of promises must enter every such region. So the finished sequence has a $1$ past spot $5$, and past spot $50$, and it differs from any single old sequence you fix in advance — because "differ from that old sequence somewhere" is itself a reachable region. The new sequence is genuinely new.

Step 5. Yet the old world names it ahead of time: the name says "read off, from whichever promises get kept, the value at each spot." Feeding the generic sequence of promises into that name yields exactly the new real.

What this tells us: finite promises plus the rule "enter every reachable region" manufacture one new infinite object that no old set equals, and a single name written in advance describes it.

Check your understanding Beginner

Formal definition Intermediate+

Work in ZFC. Fix a countable transitive model $M\models \mathrm{ZFC}$ (a ground model); the existence of such $M$ follows from the reflection theorem and downward Löwenheim-Skolem (cross-ref [42.01]) applied to enough of ZFC, and is the standard setting for the method. Ordinals, transfinite recursion, and absoluteness are as in 42.03.06; $\in$ is the only primitive.

A forcing poset is a partial order $(P,\le )$ with a greatest element $1_P$. Its elements are conditions; $q\le p$ is read "$q$ is stronger than $p$." Conditions $p,q$ are compatible if some $r$ has $r\le p$ and $r\le q$, and incompatible ($p\perp q$) otherwise. An antichain is a set of pairwise-incompatible conditions; $P$ is ccc (countable chain condition) if every antichain is countable. A set $D\subseteq P$ is dense if for every $p\in P$ there is $q\le p$ with $q\in D$, and open if it is downward closed. A set $G\subseteq P$ is a filter if it is upward closed and any two of its members are compatible within $G$ (downward directed). The filter $G$ is $M$-generic if $G\cap D\ne \varnothing$ for every dense $D\in M$; equivalently, $G$ meets every maximal antichain of $M$.

The canonical example is $\mathrm{Fn}(I,J)=\{p:p\text{ is a finite partial function }I\to J\}$, ordered by reverse inclusion ($q\le p$ iff $q\supseteq p$). The instance $\mathrm{Fn}(\omega ,2)=\mathrm{Add}(\omega ,1)$ adds a Cohen real: a generic $G$ unions to a total function $\bigcup G:\omega \to 2$ not in $M$. The instance $\mathrm{Fn}(\omega \times \omega ,2)$ adds countably many Cohen reals, and $\mathrm{Fn}(\kappa \times \omega ,2)$ adds $\kappa$ of them.

The class of $P$-names is defined by $\in$-recursion: $\tau$ is a $P$-name iff every element of $\tau$ is a pair $(\sigma ,p)$ with $\sigma$ a $P$-name and $p\in P$. Write $M^P$ for the $P$-names lying in $M$. Given a filter $G$, the interpretation (value) of a name is defined by $\in$-recursion: $$ \operatorname{val}(\tau, G) = \tau_G = {, \sigma_G : (\sigma, p) \in \tau \text{ for some } p \in G ,}. $$ The generic extension is $M[G]=\{{\tau }_G:\tau \in M^P\}$. The check name of $a\in M$ is $\check{a}=\{(\check{b},1_P):b\in a\}$, with ${\check{a}}_G=a$; the canonical name for the generic is $\dot{G}=\{(\check{p},p):p\in P\}$, with ${\dot{G}}_G=G$.

The forcing relation $p{⊩}_P\phi ({\tau }_1,\dots ,{\tau }_n)$, defined inside $M$ by recursion on names and on the formula $\phi$, holds when every $M$-generic $G\ni p$ makes $M[G]\models \phi ({\tau }_{1,G},\dots ,{\tau }_{n,G})$ — and, by the Fundamental Theorem below, this semantic description coincides with a relation $M$ can define without reference to any $G$.

Counterexamples to common slips Intermediate+

• "A generic filter is just any filter on $P$." A filter that misses even one dense set of $M$ is not generic and need not yield a model of ZFC. The defining demand is meeting every dense set of $M$; for a nontrivial (atomless) $P$ no $G\in M$ can do this, so $G\notin M$ always.

• "$M[G]$ can be built for any $G$, generic or not." The interpretation ${\tau }_G$ is defined for any $G$, but only an $M$-generic $G$ makes $M[G]\models \mathrm{ZFC}$; the proof of every axiom routes through the genericity (meeting dense sets) via the Truth Lemma.

• "$p⊩\phi$ means $\phi$ is true." Forcing is local: $p⊩\phi$ asserts that $\phi$ holds in every extension by a generic containing $p$. A weaker $p$ may force neither $\phi$ nor $\neg \phi$; only along a generic $G$ does each $\phi$ get decided, and then by some condition in $G$.

• "Names are exotic objects outside $M$." Names are ordinary sets of $M$, built by $\in$-recursion; $M^P$ is a definable subclass of $M$. The genericity, not the names, is what lives outside $M$.

Key theorem with proof Intermediate+

The organising fact is the Fundamental Theorem of Forcing: the relation "$p$ forces $\phi$," though defined by quantifying over external generic filters, is in truth internal to $M$, and along any generic $G$ the sentences true in $M[G]$ are exactly those forced by some condition in $G$. The two halves are the Definability Lemma and the Truth Lemma.

Theorem (Fundamental Theorem of Forcing). Let $M$ be a countable transitive model of ZFC and $(P,\le )\in M$ a forcing poset. For each formula $\phi (x_1,\dots ,x_n)$:

(Definability) the relation $\{(p,{\tau }_1,\dots ,{\tau }_n):p{⊩}_P\phi ({\tau }_1,\dots ,{\tau }_n)\}$ is definable in $M$ (uniformly, by a formula ${\mathrm{Frc}}_{\phi }$ with $P$ as parameter);

(Truth) for every $M$-generic $G$ and all names ${\tau }_1,\dots ,{\tau }_n\in M^P$, $$ M[G] \models \varphi(\tau_{1,G}, \dots, \tau_{n,G}) \iff (\exists p \in G)\ p \Vdash_P \varphi(\tau_1, \dots, \tau_n). $$ ^{[Kunen Ch. IV]}

Proof. Define a relation $p{⊩}^{\ast }\phi$ inside $M$ by recursion, then show it satisfies both clauses. The atomic cases $\tau \in \sigma$ and $\tau =\sigma$ are given by a simultaneous $\in$-recursion on the names: $p{⊩}^{\ast }\tau \in \sigma$ holds iff $\{q\le p:\exists (\rho ,r)\in \sigma (q\le r\wedge q{⊩}^{\ast }\tau =\rho )\}$ is dense below $p$; and $p{⊩}^{\ast }\tau =\sigma$ holds iff for every $(\rho ,r)\in \tau$ the set $\{q:q\le r\to q{⊩}^{\ast }\rho \in \sigma \}$ is dense below $p$, and symmetrically with $\tau ,\sigma$ exchanged. The recursion terminates because the name-rank strictly decreases. The connective and quantifier clauses are: $p{⊩}^{\ast }\neg \psi$ iff no $q\le p$ has $q{⊩}^{\ast }\psi$; $p{⊩}^{\ast }(\psi \wedge \chi )$ iff $p{⊩}^{\ast }\psi$ and $p{⊩}^{\ast }\chi$; $p{⊩}^{\ast }\exists x\psi (x)$ iff $\{q:\exists \tau \in M^{P}q{⊩}^{\ast }\psi (\tau )\}$ is dense below $p$. Each clause refers only to objects of $M$ (the poset, the names, density below a condition), so ${⊩}^{\ast }$ is definable in $M$ by recursion — this is the Definability Lemma, with ${\mathrm{Frc}}_{\phi }$ the resulting formula.

Two facts drive the rest. Coherence: if $p{⊩}^{\ast }\phi$ and $q\le p$ then $q{⊩}^{\ast }\phi$ (every clause is preserved by strengthening). Density of decision: for every $p$ the set $\{q\le p:q{⊩}^{\ast }\phi \text{ or }q{⊩}^{\ast }\neg \phi \}$ is dense below $p$ (if no $q\le p$ forces $\phi$, then by the $\neg$-clause $p$ itself forces $\neg \phi$). Both are proved by induction on $\phi$ alongside the recursion.

Now fix an $M$-generic $G$ and prove the Truth Lemma by induction on $\phi$, that $M[G]\models \phi ({\bar{\tau }}_G)$ iff some $p\in G$ has $p{⊩}^{\ast }\phi (\bar{\tau })$. For atomic $\phi$ the genericity of $G$ converts the dense-below-$p$ clauses into membership/equality of the interpreted names: e.g. ${\tau }_G\in {\sigma }_G$ iff some $(\rho ,r)\in \sigma$ has $r\in G$ and ${\tau }_G={\rho }_G$, which by induction and the atomic clause matches "$\exists p\in Gp{⊩}^{\ast }\tau \in \sigma$" once $G$ meets the relevant dense set (in $M$ by Definability). For $\neg \psi$: if some $p\in G$ forces $\neg \psi$ then no $q\in G$ forces $\psi$ (else a common extension in $G$ would force both, impossible by the $\neg$-clause and coherence), so by induction $M[G]/\models \psi$; conversely if no $p\in G$ forces $\psi$, then by density of decision and genericity some $p\in G$ forces $\neg \psi$. The $\wedge$ case uses that $G$ is directed; the $\exists$ case uses genericity to meet the dense set witnessing the existential by an actual name. This establishes both halves. $\square$

Bridge. The Fundamental Theorem is the foundational reason forcing controls a model it cannot see: the forcing relation is computed entirely inside $M$ (Definability), yet it predicts every truth of the external $M[G]$ (Truth), so $M$ can prove in advance what its generic extension will satisfy. This is exactly the mechanism that builds toward $M[G]\models \mathrm{ZFC}$ in the Advanced results — each axiom is verified by exhibiting, inside $M$, names and dense sets that force its instances, and the Truth Lemma then transports the forced sentence into a true one. The construction is dual to the constructible universe 42.03.06: where $L$ shrinks $V$ by admitting only definable subsets and reads its structure off an absolute recursion, forcing grows $M$ by adjoining a generic subset and reads the extension off an internal forcing relation; condensation and the Truth Lemma are the two reflection principles, one collapsing elementary submodels into levels, the other collapsing external truth into internal forcing. The central insight is that genericity converts the syntactic "dense below $p$" clauses into the semantic membership facts of $M[G]$, so meeting dense sets is the single bridge from the ground model's bookkeeping to the extension's reality. This appears again in the CH independence proof 42.03.08, where the Cohen poset's ccc plus the nice-name count force $2^{{\aleph }_0}\ge {\aleph }_2$, and in Martin's axiom and iterated forcing 42.03.09. Putting these together, the entire forcing method is one theorem — definable forcing equals generic truth — applied to ever more elaborate posets.

Exercises Intermediate+

Advanced results Master

Forcing is the outer-model technique: from a countable transitive $M\models \mathrm{ZFC}$ and a poset $P\in M$ it manufactures $M[G]$, a transitive model with the same ordinals, containing $M$ and the generic $G$, and itself a model of ZFC. The results below assemble the construction and the two structural pillars — the Fundamental Theorem and the verification of the axioms — and mark where the method's parameters (chain condition, closure) control what the extension preserves.

Theorem 1 (the generic extension; $M\subseteq M[G]$, $G\in M[G]$, leastness). For $M$-generic $G$, the class $M[G]=\{{\tau }_G:\tau \in M^P\}$ is transitive, $\mathrm{On}\cap M[G]=\mathrm{On}\cap M$, and $M\subseteq M[G]$ via the check names (${\check{a}}_G=a$) while $G={\dot{G}}_G\in M[G]$ ^{[Kunen Ch. IV]}. Moreover $M[G]$ is the least transitive model of ZF containing $M\cup \{G\}$ with the same ordinals: any such $N$ must contain ${\tau }_G$ for every $\tau \in M^P$ (the interpretation map is computed from $\tau$, $G$, and the $\in$-recursion, all available in $N$), so $M[G]\subseteq N$. Transitivity holds because if $x\in {\tau }_G$ then $x={\sigma }_G$ for some $(\sigma ,p)\in \tau$ with $p\in G$, and $\sigma \in M^P$, so $x\in M[G]$. The rank of ${\tau }_G$ is bounded by the name-rank of $\tau$, which forces $\mathrm{On}\cap M[G]=\mathrm{On}\cap M$ (no new ordinals appear; a generic adds subsets, never ordinals).

Theorem 2 (Fundamental Theorem; Definability and Truth). The forcing relation $p{⊩}_P\phi (\bar{\tau })$ is definable in $M$ uniformly in $\phi$, and $M[G]\models \phi ({\bar{\tau }}_G)$ iff $(\exists p\in G)p⊩\phi (\bar{\tau })$ ^{[Cohen 1963]}. The Key theorem gives the proof. Two corollaries organise everything downstream: the coherence $p⊩\phi ,q\le p\Rightarrow q⊩\phi$, and the decision density that below any $p$ the conditions forcing $\phi$ or forcing $\neg \phi$ are dense, so every generic decides every sentence. The relation $⊩$ being in $M$ is what lets $M$ prove statements about $M[G]$: to show $M[G]\models \psi$ it suffices to show, inside $M$, that $1_P⊩\psi$ (then every $p\in G$, in particular along $G$, forces it).

Theorem 3 ($M[G]\models \mathrm{ZFC}$). For every $M$-generic $G$, $M[G]$ satisfies every axiom of ZFC ^{[Jech Ch. 14]}. Extensionality and Foundation are inherited by transitivity; Pairing and Union are witnessed by explicit names ($\{(\sigma ,1),(\rho ,1)\}$ names $\{{\sigma }_G,{\rho }_G\}$); Infinity holds via $\check{\omega }$. Comprehension and Replacement use the Definability Lemma: to separate $\{x\in {\tau }_G:\phi (x)\}$, form the name $\{(\sigma ,p):(\sigma ,q)\in \tau ,p\le q,p⊩\phi (\sigma )\}\in M$ (the forcing predicate is definable in $M$, so this is a set of $M$), whose interpretation is the desired subset. Power Set uses the nice-name bound: a name for a subset of ${\tau }_G$ may be assembled from a single set $\{(\sigma ,p):\sigma \in \mathrm{dom}(\tau ),p\in A_{\sigma }\}$ where each $A_{\sigma }$ is an antichain, and the collection of such names is a set of $M$, so $\mathcal{P}({\tau }_G)\cap M[G]$ is the interpretation of a single name. Choice descends from a well-ordering of $M^P$ in $M$: a generic enumeration of a name's witnesses yields a choice function in $M[G]$.

Theorem 4 (preservation by chain condition and closure). If $P$ is ccc in $M$, then $M$ and $M[G]$ have the same cardinals and cofinalities ^{[Jech Ch. 14]}: by the countable-spread bound (Exercise 7), every $M[G]$-function $f:\kappa \to \lambda$ is covered by an $M$-function sending each $\alpha$ to a countable set of possible values, so no cardinal is collapsed and no cofinality changed. Dually, if $P$ is $<\kappa$-closed in $M$ (every descending $<\kappa$-sequence of conditions has a lower bound), then forcing with $P$ adds no new sequences of length $<\kappa$ of ground-model elements, so $M$ and $M[G]$ agree on $\mathcal{P}(\gamma )$ for $\gamma <\kappa$ and on cardinals up to $\kappa$. These two dials — antichains small, descending chains met — are how a forcing poset is tuned to change one cardinal invariant while freezing the rest.

Theorem 5 (ground-model definability; the cover and approximation preview). The ground model $M$ is a definable class of $M[G]$: there is a formula $\vartheta (x,r)$ and a parameter $r\in M[G]$ such that $M=\{x\in M[G]:M[G]\models \vartheta (x,r)\}$ ^{[Kunen Ch. IV]}. The proof routes through the cover and approximation properties of set forcing (Laver, Hamkins): every set of ordinals of $M[G]$ is covered by a set of $M$ of the same $M[G]$-cardinality (when $P$ has size $<\kappa$), and $M$ is approximated by its own small pieces inside $M[G]$. Consequently $M$ is not "forgotten" by its extension — the generic adds information but the ground model remains internally recoverable, the structural fact underlying set-theoretic geology and the failure of $M[G]$ to be a further forcing extension in arbitrary ways.

Synthesis. The foundational reason a model can be enlarged to settle an undecided question while keeping every axiom is the Fundamental Theorem: the forcing relation is definable in $M$, so $M$ proves in advance which sentences its generic extension satisfies, and the Truth Lemma certifies that along any generic $G$ those forced sentences are exactly the true ones. This is exactly the seam along which Cohen's independence proof runs — adjoin ${\aleph }_2^M$ Cohen reals by the ccc poset $\mathrm{Fn}({\aleph }_2\times \omega ,2)$, the countable-spread bound preserving cardinals while the nice-name count forces $2^{{\aleph }_0}\ge {\aleph }_2$, so $M[G]\models \neg \mathrm{CH}$ 42.03.08. The construction is dual to the constructible universe 42.03.06: $L$ generates constraint by admitting only definable subsets, forcing generates freedom by adjoining generic subsets; condensation collapses elementary submodels into levels and the Truth Lemma collapses external truth into internal forcing, so putting these together they are the two reflection principles that bracket every independence result. The central insight is that genericity — meeting every dense set of $M$ — converts the ground model's syntactic bookkeeping into the extension's semantic reality, and this is exactly why $M[G]\models \mathrm{ZFC}$ generalises from $M\models \mathrm{ZFC}$: each axiom's instances are forced by densely many conditions inside $M$. The chain-condition and closure dials make the method surgical, and their iteration is Martin's axiom and proper forcing 42.03.09; the ground-model definability of Theorem 5 closes the circle, so forcing is a reversible enlargement, not a destruction.

Full proof set Master

Proposition 1 (existence of generics; Rasiowa-Sikorski). If $M$ is countable and $P\in M$, then for every $p\in P$ there is an $M$-generic filter $G$ with $p\in G$.

Proof. The dense subsets of $P$ that lie in $M$ form a set $\{D_n:n\in \omega \}$ countable in $V$ (since $M$ is countable). Build a descending sequence $p=p_0\ge p_1\ge \cdots$ with $p_{n+1}\in D_n$: given $p_n$, density of $D_n$ supplies $p_{n+1}\le p_n$ in $D_n$. Let $G=\{q\in P:\exists n{p}_n\le q\}$, the upward closure of the sequence. Then $G$ is upward closed by construction; any two members $q,q^{\prime }$ have $p_n\le q$ and $p_m\le q^{\prime }$, and $p_{\max (n,m)}$ is below both, so $G$ is a filter. For genericity, each $D_n\in M$ contains $p_{n+1}\in G$, so $G\cap D_n\ne \varnothing$. Finally $p=p_0\in G$. $\square$

Proposition 2 ($⊩$ coherence and decision density). For each formula $\phi$: (i) if $p⊩\phi$ and $q\le p$ then $q⊩\phi$; (ii) $\{q\le p:q⊩\phi \text{ or }q⊩\neg \phi \}$ is dense below $p$.

Proof. Induct on $\phi$ in tandem with the recursive definition of ${⊩}^{\ast }$. Atomic: the defining clauses are "$X$ is dense below $p$" for sets $X$ that are themselves downward absolute, so strengthening $p$ to $q$ keeps $X$ dense below $q$ (a set dense below $p$ is dense below any $q\le p$), giving (i). For $\neg \psi$: $p⊩\neg \psi$ means no $r\le p$ forces $\psi$; if $q\le p$ then a fortiori no $r\le q$ forces $\psi$, so $q⊩\neg \psi$, (i). $\wedge$ and $\exists$ inherit (i) from the components. For (ii): if some $q\le p$ has $q⊩\phi$ we are done at $q$; otherwise no $r\le p$ forces $\phi$, which by the $\neg$-clause is precisely $p⊩\neg \phi$, and then every $q\le p$ forces $\neg \phi$ by (i). Either way the deciding set meets below every $p$. $\square$

Proposition 3 (the separation name; Comprehension in $M[G]$). For $\tau \in M^P$ and a formula $\phi$, the name $\rho =\{(\sigma ,p):\sigma \in \mathrm{dom}(\tau ),p⊩\sigma \in \tau \wedge \phi (\sigma )\}$ lies in $M$ and satisfies ${\rho }_G=\{x\in {\tau }_G:M[G]\models \phi (x)\}$.

Proof. $\rho \in M$: the predicate "$p⊩\sigma \in \tau \wedge \phi (\sigma )$" is definable in $M$ by the Definability Lemma, and $\mathrm{dom}(\tau )\in M$, so $\rho$ is obtained by Separation in $M$. For the value: if $x\in {\rho }_G$ then $x={\sigma }_G$ for some $(\sigma ,p)\in \rho$ with $p\in G$; then $p⊩\sigma \in \tau \wedge \phi (\sigma )$, and by the Truth Lemma $M[G]\models {\sigma }_G\in {\tau }_G\wedge \phi ({\sigma }_G)$, so $x\in {\tau }_G$ and $\phi (x)$ holds. Conversely if $x\in {\tau }_G$ and $M[G]\models \phi (x)$, write $x={\sigma }_G$ with $\sigma \in \mathrm{dom}(\tau )$; by the Truth Lemma some $p\in G$ forces $\sigma \in \tau \wedge \phi (\sigma )$, so $(\sigma ,p)\in \rho$ and $x={\sigma }_G\in {\rho }_G$. Hence ${\rho }_G$ is the separated subset, and Comprehension holds in $M[G]$. $\square$

Proposition 4 (nice names; Power Set in $M[G]$). For $\tau \in M^P$ there is a single name $\pi \in M$ with ${\pi }_G=\mathcal{P}({\tau }_G)\cap M[G]$ for every $M$-generic $G$.

Proof. Call a name $\mu$ a nice name for a subset of $\tau$ if $\mu ={\bigcup }_{\sigma \in \mathrm{dom}(\tau )}\{\sigma \}\times A_{\sigma }$ with each $A_{\sigma }$ an antichain in $P$. Every subset of ${\tau }_G$ in $M[G]$ has a nice name: if $x\subseteq {\tau }_G$ and $x={\nu }_G$, then for each $\sigma \in \mathrm{dom}(\tau )$ choose a maximal antichain $A_{\sigma }$ among conditions forcing $\sigma \in \nu$; the nice name $\mu =\bigcup \{\sigma \}\times A_{\sigma }$ has ${\mu }_G=x$ by the Truth Lemma. The class of nice names for subsets of $\tau$ is a set of $M$: it is bounded by $\mathcal{P}(\mathrm{dom}(\tau )\times P)$, which exists in $M$. Let $\Pi =\{\mu \in M:\mu \text{ is a nice name for a subset of }\tau \}$ and $\pi =\{(\mu ,1_P):\mu \in \Pi \}\in M$. Then ${\pi }_G=\{{\mu }_G:\mu \in \Pi \}=\mathcal{P}({\tau }_G)\cap M[G]$, and Power Set holds in $M[G]$ since ${\pi }_G$ is a set of $M[G]$ collecting all subsets. $\square$

Proposition 5 (ccc preserves cardinals). If $P$ is ccc in $M$ and $\kappa$ is a cardinal of $M$, then $\kappa$ is a cardinal of $M[G]$.

Proof. It suffices to preserve regular cardinals, hence to show no $M[G]$-function $f$ maps some $\lambda <\kappa$ onto $\kappa$ for $\kappa$ regular in $M$. Suppose $\dot{f}$ names such a surjection $f:\lambda \to \kappa$. By Exercise 7's bound (proved via ccc), there is $F\in M$, $F:\lambda \to [\kappa {]}^{\le \omega }$, with $f(\alpha )\in F(\alpha )$ for all $\alpha <\lambda$ in $M[G]$. Then $\mathrm{ran}(f)\subseteq {\bigcup }_{\alpha <\lambda }F(\alpha )$, a union of $\lambda$ countable sets, of cardinality $\le \lambda \cdot {\aleph }_0=\lambda <\kappa$ in $M$ (using $\kappa$ regular and $\lambda <\kappa$). This contradicts $f$ being onto $\kappa$. Hence no such surjection exists and $\kappa$ remains a cardinal in $M[G]$. The same covering argument preserves cofinalities. $\square$

Proposition 6 (the Cohen real is new and unbounded). Let $c=\bigcup G:\omega \to 2$ for $G$ generic over $\mathrm{Fn}(\omega ,2)$. Then $c\notin M$, and $c$ differs from every $g:\omega \to 2$ in $M$.

Proof. By Exercise 3, $c$ is a total function $\omega \to 2$. Fix $g\in M$, $g:\omega \to 2$, and let $E_g=\{p\in \mathrm{Fn}(\omega ,2):\exists n(n\in \mathrm{dom}(p)\wedge p(n)\ne g(n))\}$. Then $E_g\in M$ and $E_g$ is dense: any condition $p$ has a free coordinate $n\notin \mathrm{dom}(p)$ (domains are finite), and $p\cup \{(n,1-g(n))\}\le p$ lies in $E_g$. Genericity gives $q\in G\cap E_g$, so $c(n)=q(n)\ne g(n)$ at the witnessing $n$, hence $c\ne g$. As $g\in M$ was arbitrary, $c\notin M$. (Equivalently $\mathrm{Fn}(\omega ,2)$ is atomless, so Exercise 5 applies.) $\square$

Connections Master

• The constructible universe $L$ and the consistency of GCH 42.03.06 is the direct prerequisite and the dual technique. That unit shrinks $V$ to the least inner model by admitting only definable subsets, reading every structural fact off the absolute constructible recursion and condensation, and proves $L\models \mathrm{GCH}$. This unit grows a ground model by adjoining a generic subset, reading the extension off the internal forcing relation and the Truth Lemma. Condensation (collapse an elementary submodel of $L_{\alpha }$ into a level) and the Fundamental Theorem (collapse external truth in $M[G]$ into internal forcing in $M$) are the two reflection engines; together they bracket the continuum from below ($L$: GCH consistent) and above (forcing: $\neg$CH consistent).

• The independence of the continuum hypothesis 42.03.08 is the co-produced payoff that runs this machinery on a specific poset: forcing with $\mathrm{Fn}({\aleph }_2^M\times \omega ,2)$ adds ${\aleph }_2$ Cohen reals, the ccc preservation (Theorem 4, Proposition 5) keeps every cardinal, and the nice-name count (Proposition 4) bounds and then realises $2^{{\aleph }_0}={\aleph }_2$ in $M[G]$, giving $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{ZFC}+\neg \mathrm{CH})$. The Axiom of Choice 42.03.05 enters as the descent of Choice to $M[G]$ in Theorem 3 (a well-order of $M^P$ in $M$ yields choice in the extension), the symmetric-extension variant of which produces models of $\neg \mathrm{AC}$.

• Martin's axiom and iterated forcing 42.03.09 is the co-produced continuation that iterates the single-step forcing of this unit. Where one Cohen poset adds one batch of reals, a finite-support ccc iteration of length ${\aleph }_2$ forces Martin's axiom plus $2^{{\aleph }_0}={\aleph }_2$, settling the Suslin and Whitehead problems; the two-step iteration $P\ast \dot{Q}$ and the preservation of ccc under iteration are exactly the chain-condition and name machinery developed here. First-order definability and absoluteness [42.01] supply the countable transitive ground model (via Löwenheim-Skolem and reflection) and the Lévy-hierarchy absoluteness that makes the forcing relation $M$-definable.

Historical & philosophical context Master

Paul Cohen introduced forcing in two 1963-1964 notes to the Proceedings of the National Academy of Sciences and the 1966 monograph Set Theory and the Continuum Hypothesis ^{[Cohen 1963]}, constructing from a countable standard model of ZF an extension in which the continuum hypothesis fails. His original presentation defined a condition as a finite consistent set of forcing statements about the new sets and defined "$P$ forces a sentence" by recursion on logical form so that the relation is expressible in the ground model; the truth lemma — that for a complete sequence of conditions the forced sentences are exactly the true ones — was the technical heart. Together with Gödel's 1938 proof that $L\models \mathrm{GCH}$, Cohen's result established the full independence of CH from ZFC, the problem first on Hilbert's 1900 list; Cohen received the Fields Medal in 1966.

The method was reorganised almost immediately. Dana Scott and Robert Solovay recast forcing as the theory of Boolean-valued models $V^B$, replacing the syntactic forcing relation with a Boolean truth value $[[\phi ]]$ in the completion $B=\mathrm{RO}(P)$ of the poset, an approach developed in Scott's lectures and in Solovay's work of the mid-1960s; Joseph Shoenfield gave the clean poset-and-dense-set formulation now standard in Kunen and Jech ^{[Kunen Ch. IV]}. Azriel Lévy and Solovay used forcing to show large-cardinal axioms cannot decide CH, and Solovay's 1970 model (Lebesgue measurability of all sets from an inaccessible) and the Lévy collapse showed forcing's reach beyond CH. Subsequent refinements — Martin and Solovay's Martin's axiom, the proper-forcing axiom of Shelah, and Hugh Woodin's ${\mathbb{P}}_{\max }$ — made iterated and class forcing the central instrument of modern set theory, with Joel Hamkins and Jonas Reitz developing set-theoretic geology from the ground-model definability that closes this unit.

Bibliography Master

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  publisher = {College Publications},
  year      = {2011}
}

@book{jech2003set,
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}

@article{cohen1963independence,
  author  = {Cohen, Paul J.},
  title   = {The independence of the continuum hypothesis},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {50},
  year    = {1963},
  pages   = {1143--1148}
}

@article{cohen1964independence2,
  author  = {Cohen, Paul J.},
  title   = {The independence of the continuum hypothesis, II},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {51},
  year    = {1964},
  pages   = {105--110}
}

@book{cohen1966set,
  author    = {Cohen, Paul J.},
  title     = {Set Theory and the Continuum Hypothesis},
  publisher = {W. A. Benjamin},
  year      = {1966}
}

@article{solovay1970measurable,
  author  = {Solovay, Robert M.},
  title   = {A model of set-theory in which every set of reals is Lebesgue measurable},
  journal = {Annals of Mathematics},
  volume  = {92},
  year    = {1970},
  pages   = {1--56}
}

@incollection{scott1967proof,
  author    = {Scott, Dana},
  title     = {A proof of the independence of the continuum hypothesis},
  booktitle = {Mathematical Systems Theory},
  volume    = {1},
  year      = {1967},
  pages     = {89--111}
}

@article{shoenfield1971unramified,
  author  = {Shoenfield, Joseph R.},
  title   = {Unramified forcing},
  journal = {Proceedings of Symposia in Pure Mathematics},
  volume  = {13},
  year    = {1971},
  pages   = {357--381}
}