42.03.08 · mathematical-logic / set-theory-forcing

Forcing II: Cohen and the Independence of the Continuum Hypothesis

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Anchor (Master): Kunen 2011 *Set Theory* (College Publications) Ch. V and VIII in full (the Δ-system lemma for the ccc of finite-partial-function posets, the cardinal-preservation theorem, the nice-name bound 2^{ℵ₀} ≤ |Fn(κ×ω,2)|^{ℵ₀} = κ^{ℵ₀}, the Cohen model for ¬CH, the countably-closed collapse forcing CH/GCH, and the independence of ◊ and the Suslin hypothesis); Jech 2003 *Set Theory* 3e (Springer) Ch. 15, 28-29 (the continuum function under forcing, Easton's theorem on the regulars, and cardinal invariants of the continuum); Bartoszyński-Judah 1995 *Set Theory: On the Structure of the Real Line* (A K Peters) for the Cichoń diagram and the independence of the cardinal invariants

Intuition Beginner

The continuum hypothesis asks a single, ancient question: is there a size of infinity strictly between the counting numbers and the real line? Cantor believed the answer was no — that the reals are the very next infinity after the whole numbers — but he could neither prove it nor refute it, and the question became the first problem on Hilbert's famous 1900 list. The surprise, settled only in 1963, is that the usual rules of mathematics simply do not decide it. You can obey every axiom and still go either way.

The previous unit built the machine that proves this. It showed how to start with a fixed world of sets and grow it by adding new objects made from tiny finite promises, while keeping every axiom intact. This unit runs that machine to its payoff. The trick is to add not one new infinite sequence of $0$ s and $1$ s but a huge crowd of them — as many as the second uncountable size of infinity — each one genuinely new and different from all the others.

When you add that many new reals, the real line is forced to be at least that big, so the next-size-up answer becomes false in the grown world. And because the promises are finite, the crowd is added gently enough that no old size of infinity gets squashed or merged. The result is a perfectly lawful world of mathematics where the continuum hypothesis fails. Pair it with the slim world from two units back, where it holds, and the question stands revealed as undecidable.

Visual Beginner

To make the continuum large, add a whole grid of new bits: one row for each of many new reals, one column for each spot along a real. A finite promise fills in only finitely many cells of this grid; a generic crowd of promises fills every cell, and each row reads off as a brand-new real.

        spot 0  spot 1  spot 2  spot 3 ...   (columns: positions in a real)
real 0:   1       .       0       .          a finite promise fills
real 1:   .       0       .       1          only a few scattered cells
real 2:   0       .       .       .
  .
  .       (there is one row for EACH of the many new reals)

  a finite condition = finitely many filled cells anywhere in the grid
  a generic crowd     = every cell eventually filled
  each finished ROW   = one brand-new real, all rows distinct

The table below contrasts the two famous worlds. The slim world admits only describable sets and the next-size answer comes out true; the grown world adds a crowd of new reals and the answer comes out false. Same axioms in both.

feature	slim world (from 42.03.06)	grown world (this unit)
how it is built	keep only describable sets	add a crowd of new reals
number of reals	the next size up	the second size up (or more)
the continuum hypothesis	holds	fails
every axiom of set theory	holds	holds

The key idea: adding the second uncountable size of new reals by finite promises makes the real line at least that big, while finiteness keeps every old infinity intact — so the continuum hypothesis fails lawfully.

Worked example Beginner

Let us add two new reals at once and watch them come out different — the same idea that, scaled up to a huge crowd, defeats the continuum hypothesis. A condition is now a finite list of filled cells in a two-row grid: it fixes a few bits of real $A$ and a few bits of real $B$ .

Step 1. Start with the empty promise: no cell filled. It is the weakest condition.

Step 2. A stronger condition fills more cells, never erasing one. For instance "row $A$ spot $0$ is $1$ ; row $B$ spot $0$ is $0$ " is a condition fixing two cells. Adding "row $A$ spot $1$ is $1$ " makes it stronger.

Step 3. Consider the demand " $A$ and $B$ differ somewhere." The conditions settling it form a region you can always reach: from any finite promise, find a spot $n$ left blank in both rows and fill row $A$ at $n$ with $1$ and row $B$ at $n$ with $0$ . Since each promise fills only finitely many cells, such a blank spot always exists, so this region is dense.

Step 4. A generic crowd of promises must enter that region, so the finished reals $A$ and $B$ differ at some spot. They are two distinct new reals.

Step 5. Now picture the same set-up with a row for each member of a very large index set instead of two rows. The matching demand "rows $ξ$ and $η$ differ somewhere" is dense for every pair, so all the finished reals are pairwise distinct. The real line must hold them all.

What this tells us: finite promises plus the rule "enter every reachable region" manufacture as many pairwise-distinct new reals as you have rows, and that forces the continuum to be at least that large.

Check your understanding Beginner

Exercise (easy, multiple-choice).

Why does adding a large crowd of new reals make the continuum hypothesis fail?

Because the new reals are all equal to old reals.
Because the crowd is pairwise distinct, so the real line must be at least as big as the crowd, overshooting the next size up.
Because adding reals removes old reals.
Because the new reals are finite.

Hint

Count how many distinct new reals the real line is now forced to contain.

Answer

B. The crowd is pairwise distinct, so the grown real line contains at least that many reals; choosing a crowd of the second uncountable size makes the continuum at least that big, past the next-size-up answer. Feedback-correct: more distinct reals means a larger continuum. Feedback-wrong: the new reals are genuinely new and infinite, and no old real is lost (A, C, D are false).

Exercise (easy, multiple-choice).

Why does adding the crowd by finite promises keep every old size of infinity intact?

Because finite promises can merge two sizes into one.
Because the gentle, finite-promise structure stops any old size from being squashed down to a smaller one.
Because finite promises delete reals.
Because there are no old sizes of infinity.

Hint

The danger would be an old large size getting matched up with a smaller one; finiteness blocks that.

Answer

B. The finite-promise structure keeps the forcing gentle enough that no old size of infinity is collapsed onto a smaller one, so the sizes that bounded the crowd still stand. Feedback-correct: preserving the old sizes is what lets the count "second size up" stay meaningful. Feedback-wrong: finite promises neither merge sizes nor delete reals (A, C are false), and old sizes certainly exist (D is false).

Formal definition Intermediate+

Work in ZFC with a countable transitive model $M ⊨ ZFC$ and the forcing apparatus of 42.03.07: a forcing poset $(P, \leq, 1_{P})$ , conditions, compatibility, antichains, dense sets, $M$ -generic filters $G$ , the class $M^{P}$ of $P$ -names, the interpretation $τ_{G}$ , the generic extension $M [G]$ , and the forcing relation $p ⊩ φ$ with its Fundamental Theorem (Definability and Truth).

A poset $P$ satisfies the countable chain condition (ccc) if every antichain $A \subseteq P$ is countable. For an index set $I$ and a set $J$ , the Cohen poset $Fn (I, J)$ is the set of finite partial functions $I \to J$ ordered by reverse inclusion. The instance that adds $κ$ Cohen reals is $$ \operatorname{Add}(\omega, \kappa) = \operatorname{Fn}(\kappa \times \omega, 2), $$ whose generic $G$ yields, for each $ξ < κ$ , a Cohen real $c_{ξ} \in 2^{ω}$ with $c_{ξ} (n) = (⋃ G) (ξ, n)$ ; distinct indices give distinct reals.

A family $D$ of finite sets is a $Δ$ -system (sunflower) with root $r$ if $a \cap b = r$ for all distinct $a, b \in D$ . The continuum function is $κ \mapsto 2^{κ}$ ; the continuum hypothesis (CH) is $2^{ℵ_{0}} = ℵ_{1}$ , and the generalised continuum hypothesis (GCH) is $2^{ℵ_{α}} = ℵ_{α + 1}$ for all $α$ .

A name $μ$ is a nice name for a subset of $\overset{ω}{ˇ}$ if $μ = ⋃_{n \in ω} {\overset{n}{ˇ}} \times A_{n}$ with each $A_{n}$ an antichain of $P$ . Every subset of $ω$ in $M [G]$ has a nice name (cross-ref 42.03.07, Proposition 4 there), so counting nice names in $M$ bounds $2^{ℵ_{0}}$ in $M [G]$ from above.

Counterexamples to common slips Intermediate+

"Adding $ℵ_{2}$ reals automatically makes $2^{ℵ_{0}} = ℵ_{2}$ ." It makes $2^{ℵ_{0}} \geq ℵ_{2}$ at once, but the equality needs the upper bound from the nice-name count; without it the continuum could be larger. Both halves — a lower bound from distinct generics, an upper bound from counting names — are required.
"Any forcing that adds reals preserves cardinals." Only forcing with a chain condition (or closure) does. The collapse poset $Fn (ω, ω_{1})$ adds a surjection $ω \to ω_{1}^{M}$ and is not ccc; it collapses $ω_{1}$ . The Cohen poset is ccc precisely because of the $Δ$ -system lemma.
"ccc is about chains." The name is historical; ccc is the antichain condition — every antichain is countable. (In a Boolean algebra the two notions coincide via the regular-open completion, which is the source of the name.)
"The continuum can be any cardinal." ZFC proves $cf (2^{ℵ_{0}}) > ℵ_{0}$ (König), so $2^{ℵ_{0}} \neq = ℵ_{ω}$ ; the Cohen poset can force $2^{ℵ_{0}} = κ$ only for $κ$ of uncountable cofinality with $κ^{ℵ_{0}} = κ$ .

Key theorem with proof Intermediate+

The organising fact is that the Cohen poset is ccc, and ccc forcing preserves cardinals; everything in the independence proof rests on this pair. The chain condition is delivered by the $Δ$ -system lemma, a purely combinatorial statement about finite sets.

Theorem (the Cohen poset is ccc). For every index set $I$ the poset $Fn (I, 2)$ satisfies the countable chain condition. ^{[Kunen Ch. V]}

Proof. Suppose toward a contradiction that $A = {p_{α} : α < ω_{1}}$ is an antichain of $Fn (I, 2)$ of size $ℵ_{1}$ (an uncountable antichain contains one of size $ℵ_{1}$ ). Each $p_{α}$ has finite domain $d_{α} = dom (p_{α}) \in [I]^{< ω}$ . Apply the $Δ$ -system lemma (proved below) to ${d_{α} : α < ω_{1}}$ : there is an uncountable $W \subseteq ω_{1}$ such that ${d_{α} : α \in W}$ is a $Δ$ -system with some finite root $r \subseteq I$ . For $α \in W$ the restriction $p_{α} ↾ r$ is a function $r \to 2$ , and there are only $2^{∣ r ∣}$ such functions — finitely many. Since $W$ is uncountable, two indices $α \neq = β$ in $W$ have $p_{α} ↾ r = p_{β} ↾ r$ . Outside $r$ the domains $d_{α}$ and $d_{β}$ are disjoint (the $Δ$ -system property), so $p_{α}$ and $p_{β}$ agree on $d_{α} \cap d_{β} = r$ and conflict nowhere; hence $p_{α} \cup p_{β}$ is a finite partial function, a common extension, and $p_{α}, p_{β}$ are compatible. That contradicts $A$ being an antichain. Therefore no uncountable antichain exists and $Fn (I, 2)$ is ccc. $□$

Lemma ( $Δ$ -system lemma). Every uncountable family $F$ of finite sets has an uncountable subfamily forming a $Δ$ -system. ^{[Kunen Ch. V]}

Proof. Thin $F$ to size $ℵ_{1}$ . Since each member is finite, by the pigeonhole on $ℵ_{1}$ there is a fixed $n$ with uncountably many members of size exactly $n$ ; replace $F$ by these and induct on $n$ . For $n = 0$ all members are empty, a $Δ$ -system with root $\emptyset$ . For the step, fix $ℵ_{1}$ -many sets of size $n$ . If some element $x$ lies in uncountably many of them, restrict to those and remove $x$ , reducing to size $n - 1$ , get a $Δ$ -system with root $s$ , and add $x$ back for root $s \cup {x}$ . Otherwise each element lies in only countably many members; then a transfinite recursion of length $ω_{1}$ chooses pairwise-disjoint members (at each step the previously chosen finitely-supported $< ℵ_{1}$ sets meet only countably many others, leaving one disjoint from all), giving a $Δ$ -system with root $\emptyset$ . $□$

Bridge. The countable chain condition is the foundational reason a vast crowd of Cohen reals can be added without disturbing the cardinal structure: the $Δ$ -system lemma converts the finiteness of conditions into the smallness of antichains, and smallness of antichains is exactly what the countable-cover bound of 42.03.07 needs to certify that no $M [G]$ -function collapses a cardinal. This is exactly the mechanism that builds toward Cohen's theorem in the Advanced results — once $Fn (ℵ_{2} \times ω, 2)$ is ccc, every cardinal of $M$ survives into $M [G]$ , so the $ℵ_{2}$ distinct generic reals stay $ℵ_{2}$ -many and $2^{ℵ_{0}} \geq ℵ_{2}$ there. The chain condition generalises the closure dial of 42.03.07 from the other side: $< κ$ -closure freezes the small sequences, ccc freezes the cardinals, and the two together let a poset be tuned to move one invariant while pinning the rest. The central insight is that preservation is a counting phenomenon — antichains deciding a value are antichains, hence countable, hence the value ranges over a ground-model-countable set — so the same nice-name bookkeeping that proved Power Set in $M [G]$ now bounds the continuum. This appears again in the CH-forcing direction, where countably-closed forcing adds no reals and so raises nothing, and in the Suslin and $◊$ independence and in Martin's axiom and iterated forcing 42.03.09. Putting these together, cardinal arithmetic in a forcing extension is governed by two dials, the chain condition and the closure, and Cohen's theorem is the first turn of the first dial.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that for $Add (ω, κ) = Fn (κ \times ω, 2)$ and any $ξ \neq = η$ in $κ$ , the set $E_{ξ, η} = {p : \exists n (ξ, n), (η, n) \in dom (p) \land p (ξ, n) \neq = p (η, n)}$ is dense. Conclude the generic reals $c_{ξ}, c_{η}$ are distinct.

Hint

Any finite condition leaves some column $n$ blank in both rows $ξ$ and $η$ ; fill it with opposite bits.

Answer

Given $p \in Add (ω, κ)$ , its domain is finite, so some $n$ has neither $(ξ, n)$ nor $(η, n)$ in $dom (p)$ . Let $q = p \cup {((ξ, n), 0), ((η, n), 1)}$ . Then $q$ is a finite partial function, $q \supseteq p$ so $q \leq p$ , and $q (ξ, n) = 0 \neq = 1 = q (η, n)$ , so $q \in E_{ξ, η}$ . Hence every condition extends into $E_{ξ, η}$ and it is dense. Each $E_{ξ, η} \in M$ (definable from $ξ, η$ ), so an $M$ -generic $G$ meets it: some $q \in G$ has $c_{ξ} (n) = q (ξ, n) \neq = q (η, n) = c_{η} (n)$ , whence $c_{ξ} \neq = c_{η}$ . Rubric: full credit for the blank-column extension, density, and the distinctness from meeting $E_{ξ, η}$ .

Exercise 4 (medium, symbolic).

Using the nice-name bound, show that if $M ⊨ GCH$ and $P = Fn (κ \times ω, 2)$ with $κ^{ℵ_{0}} = κ$ in $M$ , then $M [G] ⊨ 2^{ℵ_{0}} \leq κ$ .

Hint

Count nice names for subsets of $\overset{ω}{ˇ}$ : each is determined by a function $ω \to (antichains of P)$ , and antichains are countable by ccc.

Answer

Every real of $M [G]$ has a nice name $μ = ⋃_{n} {\overset{n}{ˇ}} \times A_{n}$ with each $A_{n}$ a (countable, by ccc) antichain of $P$ . Now $∣ P ∣ = ∣ [κ \times ω]^{< ω} \cdot 2∣ = κ$ in $M$ , so the number of countable antichains is at most $∣ P ∣^{ℵ_{0}} = κ^{ℵ_{0}} = κ$ . A nice name is an $ω$ -indexed sequence of such antichains, so there are at most $κ^{ℵ_{0}} = κ$ nice names. Distinct reals of $M [G]$ have distinct nice names, so $(2^{ℵ_{0}})^{M [G]} \leq κ$ , computed in $M$ where $κ$ is a cardinal preserved by ccc. Rubric: full credit for the nice-name representation, the count $∣ P ∣ = κ$ , the bound $κ^{ℵ_{0}} = κ$ , and the transfer to $M [G]$ .

Exercise 5 (medium, symbolic).

Prove the $n = 1$ base case of the $Δ$ -system step concretely: an uncountable family of singletons ${x_{α}}$ contains an uncountable $Δ$ -system. Identify its root.

Hint

Either one value repeats uncountably often, or all values are distinct.

Answer

Let $F = {{x_{α}} : α < ω_{1}}$ . Case 1: some value $x$ has ${x_{α}} = {x}$ for uncountably many $α$ ; these form a $Δ$ -system with root ${x}$ (all members equal). Case 2: each value occurs only countably often; then choose $ω_{1}$ -many $α$ with pairwise-distinct $x_{α}$ (a transfinite choice, possible since at each stage only countably many indices are excluded), giving pairwise-disjoint singletons, a $Δ$ -system with root $\emptyset$ . Either way an uncountable $Δ$ -system results. Rubric: full credit for both cases and the correctly identified root in each.

Exercise 6 (hard, symbolic).

Prove that $Fn (ω, ω_{1})$ (finite partial functions $ω \to ω_{1}$ ) is not ccc, and that forcing with it collapses $ω_{1}$ .

Hint

Exhibit an uncountable antichain among single-point conditions, and show $⋃ G$ is a surjection $ω \to ω_{1}^{M}$ .

Answer

For each $β < ω_{1}$ let $p_{β} = {(0, β)}$ . Any two $p_{β}, p_{β^{'}}$ with $β \neq = β^{'}$ assign different values to $0$ , so they have no common extension: they are incompatible. Thus ${p_{β} : β < ω_{1}}$ is an antichain of size $ℵ_{1}$ , and the poset is not ccc. For collapse: for each $β < ω_{1}$ and $n < ω$ the set $D_{β} = {p : β \in ran (p)}$ is dense (extend any $p$ by sending a fresh point of $ω$ to $β$ ), and $D_{β} \in M$ . A generic $G$ meets every $D_{β}$ , so $f = ⋃ G : ω \to ω_{1}^{M}$ is a total surjection in $M [G]$ (it is single-valued as $G$ is a filter, total by the sets ${p : n \in dom p}$ , onto by the $D_{β}$ ). Hence $ω_{1}^{M}$ is countable in $M [G]$ and is collapsed. Rubric: full credit for the antichain, the density of $D_{β}$ , and the surjection witnessing the collapse.

Exercise 7 (hard, symbolic).

Assume $M ⊨ GCH$ . Show that forcing with the countably-closed poset $Add (ω_{1}, 1) = Fn_{ω_{1}} (ω_{1}, 2)$ of countable partial functions adds no new real, and explain why this is the engine behind forcing CH.

Hint

A descending $ω$ -sequence of countable conditions has a lower bound (their union); use this to meet, inside $M$ , every name for a real's value.

Answer

The poset is $< ω_{1}$ -closed: any descending sequence $p_{0} \geq p_{1} \geq \dots$ of length $ω$ has lower bound $⋃_{n} p_{n}$ , a countable partial function. Let $\overset{c}{˙}$ name a function $\overset{ω}{ˇ} \to \overset{ˇ}{2}$ and fix $p$ . Build a descending sequence $p = p_{0} \geq p_{1} \geq \dots$ with $p_{n + 1}$ deciding $\overset{c}{˙} (\overset{n}{ˇ})$ (possible by density of decision). By closure their union $q$ is a condition below all $p_{n}$ , and $q$ decides $\overset{c}{˙} (\overset{n}{ˇ})$ for every $n$ , hence $q$ determines a ground-model function $g \in M$ with $q ⊩ \overset{c}{˙} = \overset{g}{ˇ}$ . So no condition forces $\overset{c}{˙}$ to be new: every real of $M [G]$ already lies in $M$ . Consequently the old reals are all the reals, and if the forcing also well-orders them in type $ω_{1}$ (e.g. by coding a bijection $ω_{1} \to (2^{ω})^{M}$ ), then $M [G] ⊨ 2^{ℵ_{0}} = ℵ_{1}$ , i.e. CH. Rubric: full credit for closure giving the lower bound, the no-new-reals capture argument, and the link to CH.

Advanced results Master

The continuum problem is settled by running the machine of 42.03.07 on Cohen's poset in one direction and on a countably-closed poset in the other. The results below assemble the full independence proof, then mark the further independence phenomena the same method reaches.

Theorem 1 (ccc preserves cardinals and cofinalities). If $P$ is ccc in $M$ , then $M$ and $M [G]$ have the same cardinals and the same cofinality function ^{[Jech Ch. 15]}. By the maximal-antichain bound, any $\dot{f}$ forced to be $f : κ \to λ$ in $M [G]$ admits $F : κ \to [λ]^{\leq ω}$ in $M$ with $f (α) \in F (α)$ for all $α$ : a maximal antichain $A_{α} \in M$ deciding $\dot{f} (\overset{α}{ˇ})$ is countable by ccc, so $F (α) = {m : \exists q \in A_{α}, q ⊩ \dot{f} (\overset{α}{ˇ}) = \overset{m}{ˇ}}$ is countable. Hence $ran (f) \subseteq ⋃_{α} F (α)$ has $M$ -cardinality $\leq ∣ κ ∣ \cdot ℵ_{0}$ , so no regular cardinal is mapped onto by a shorter sequence: no cardinal collapses and no cofinality changes.

Theorem 2 (Cohen's theorem; $Con (ZFC) \Rightarrow Con (ZFC + \neg CH)$ ). Let $M ⊨ ZFC + GCH$ be countable transitive and force with $P = Fn (ℵ_{2}^{M} \times ω, 2) = Add (ω, ℵ_{2})$ ^{[Cohen 1963]}. Then $P$ is ccc (Key theorem), so all cardinals are preserved (Theorem 1); in particular $ℵ_{1}^{M} = ℵ_{1}^{M [G]}$ and $ℵ_{2}^{M} = ℵ_{2}^{M [G]}$ . The $ℵ_{2}$ generic reals $c_{ξ}$ are pairwise distinct (the diagonal-difference sets are dense), so $M [G] ⊨ 2^{ℵ_{0}} \geq ℵ_{2}$ . Conversely $∣ P ∣ = ℵ_{2}$ in $M$ and $GCH$ gives $ℵ_{2}^{ℵ_{0}} = ℵ_{2}$ , so the nice-name count yields $M [G] ⊨ 2^{ℵ_{0}} \leq ℵ_{2}$ . Hence $M [G] ⊨ 2^{ℵ_{0}} = ℵ_{2} > ℵ_{1}$ , so $M [G] ⊨ \neg CH$ . The hypothesis $Con (ZFC)$ supplies (via the reflection theorem and Löwenheim-Skolem, [42.01]) such a countable transitive $M$ , and $M [G] ⊨ ZFC$ by 42.03.07; therefore $Con (ZFC + \neg CH)$ .

Theorem 3 (forcing CH and GCH; the other half of independence). Countably-closed forcing adds no new reals, and a $< κ$ -closed collapse forces CH (Exercise 7) ^{[Kunen Ch. V]}. Equivalently, in a model where $2^{ℵ_{0}} > ℵ_{1}$ , the poset $Coll (ω_{1}, 2^{ℵ_{0}})$ of countable approximations to a surjection $ω_{1} \to (2^{ω})$ adds no real, collapses $2^{ℵ_{0}}$ to $ℵ_{1}$ , and so forces $2^{ℵ_{0}} = ℵ_{1}$ , i.e. CH; iterating a closed collapse on every cardinal forces GCH. Together with Gödel's $L ⊨ GCH$ (42.03.06) this gives $Con (ZFC) \Rightarrow Con (ZFC + CH)$ — indeed $Con (ZFC + GCH)$ — by either route. Combining Theorems 2 and 3 yields the independence of CH: ZFC proves neither CH nor $\neg$ CH, settling Hilbert's first problem.

Theorem 4 (Easton's theorem; the continuum function on the regulars). ZFC proves of the continuum function on regular cardinals only monotonicity $κ \leq λ \Rightarrow 2^{κ} \leq 2^{λ}$ and König's inequality $cf (2^{κ}) > κ$ . For any class function $E$ on the regular cardinals with $E (κ) \leq E (λ)$ for $κ \leq λ$ and $cf (E (κ)) > κ$ , there is a class forcing (Easton-support product of the Cohen posets $Add (κ, E (κ))$ ) making $2^{κ} = E (κ)$ for all regular $κ$ ^{[Jech Ch. 15]}. Thus the continuum function on the regulars is unconstrained by ZFC beyond those two laws, sharpening the cofinality and cardinal-exponentiation analysis of 42.03.04; on singulars the behaviour is governed instead by the Singular Cardinals Hypothesis and pcf theory, where ZFC proves far more.

Theorem 5 (further independence by forcing). The Suslin hypothesis is independent: $◊$ (which holds in $L$ and after the countably-closed collapse) yields a Suslin tree, while a finite-support ccc iteration of length $ℵ_{2}$ forces Martin's axiom and refutes every Suslin tree ^{[Jech Ch. 15]}. The combinatorial principle $◊$ and the Borel conjecture, the Kurepa hypothesis, and the cardinal invariants of the continuum (the bounding and dominating numbers $b, d$ , the additivity and cofinality of measure and category arranged in the Cichoń diagram) are all settled only by forcing: ZFC proves the inequalities the diagram records, and adding Cohen, random, dominating, or Sacks reals separates the rest. Each such result instantiates the same template — choose a poset whose chain condition or closure preserves the cardinals you need, and whose generic realises the configuration you want.

Synthesis. The foundational reason the continuum hypothesis is neither provable nor refutable is that the same forcing machine runs both ways: Cohen's ccc poset $Add (ω, ℵ_{2})$ adds $ℵ_{2}$ distinct reals while the $Δ$ -system lemma keeps every cardinal, forcing $2^{ℵ_{0}} = ℵ_{2}$ and $\neg$ CH, and the countably-closed collapse adds no reals while shrinking the continuum to $ℵ_{1}$ , forcing CH. This is exactly the seam predicted by the Fundamental Theorem of 42.03.07: the forcing relation is definable in $M$ , so $M$ proves in advance which continuum value each poset delivers. The construction is dual to the constructible universe 42.03.06: $L$ generates constraint and proves CH from condensation, forcing generates freedom and refutes CH from genericity; condensation and the nice-name count are the two reflection principles, one bounding the continuum below at $ℵ_{1}$ , the other lifting it above to $ℵ_{2}$ , and putting these together they bracket Hilbert's first problem from both sides. The central insight is that cardinal arithmetic in an extension is a counting phenomenon governed by two dials — the chain condition freezes cardinals, the closure freezes small sequences — so the continuum function is as free as Easton's theorem 42.03.04 allows on the regulars and no freer. This is the foundational reason large cardinals cannot decide CH (Levy-Solovay: small forcing is absorbed), and the bridge is iteration: a finite-support ccc iteration forces Martin's axiom and the independence of the Suslin hypothesis 42.03.09, generalising the single Cohen step into the modern instrument that settles the cardinal invariants of the continuum.

Full proof set Master

Proposition 1 ( $Δ$ -system lemma, full statement). Let $κ$ be regular and uncountable and let $F$ be a family of finite sets with $∣ F ∣ = κ$ . Then $F$ has a subfamily of size $κ$ forming a $Δ$ -system.

Proof. By pigeonhole on the regular $κ$ , some fixed $n$ has $κ$ -many members of size $n$ ; restrict $F$ to these and induct on $n$ . Base $n = 0$ : $κ$ empty sets, root $\emptyset$ . Step: given $κ$ -many sets of size $n + 1$ , consider whether some point $x$ lies in $κ$ -many of them. If so, restrict to those, delete $x$ to get $κ$ -many sets of size $n$ , find by induction a $Δ$ -system of size $κ$ with root $s$ , and restore $x$ for root $s \cup {x}$ . If no point lies in $κ$ -many members, build a $Δ$ -system with root $\emptyset$ by transfinite recursion: having chosen $< κ$ pairwise-disjoint members, their union is a set of size $< κ$ meeting (by hypothesis) only $< κ$ members, so by regularity of $κ$ some member is disjoint from all chosen ones; continue $κ$ steps. $□$

Proposition 2 ( $Fn (I, 2)$ is ccc). For every $I$ , every antichain of $Fn (I, 2)$ is countable.

Proof. If $A$ were an antichain of size $ℵ_{1}$ , apply Proposition 1 with $κ = ℵ_{1}$ to the domains: an $ℵ_{1}$ -sized $Δ$ -system with finite root $r$ . The $2^{∣ r ∣}$ restrictions to $r$ are finite in number, so two conditions in the system agree on $r$ ; off $r$ their domains are disjoint, so they are compatible, contradicting antichain-hood. Hence $∣ A ∣ \leq ℵ_{0}$ . $□$

Proposition 3 (ccc cardinal preservation). If $P$ is ccc in $M$ and $κ$ is a cardinal of $M$ , then $κ$ is a cardinal of $M [G]$ , and $cf^{M} (κ) = cf^{M [G]} (κ)$ .

Proof. Preserving cardinals reduces to preserving regulars and reduces further to: no $f : λ \to κ$ ( $λ < κ$ , $κ$ regular in $M$ ) of $M [G]$ is onto. Suppose $\dot{f}$ names such an $f$ . For each $α < λ$ pick in $M$ a maximal antichain $A_{α}$ of conditions deciding $\dot{f} (\overset{α}{ˇ})$ ; by ccc each $A_{α}$ is countable, so $F (α) = {m : \exists q \in A_{α}, q ⊩ \dot{f} (\overset{α}{ˇ}) = \overset{m}{ˇ}}$ is countable. Then in $M [G]$ , $f (α) \in F (α)$ (a $p \in G$ forces the value, and $p$ is compatible with the deciding $q \in A_{α}$ ), so $ran (f) \subseteq ⋃_{α < λ} F (α)$ , of $M$ -cardinality $\leq λ \cdot ℵ_{0} = λ < κ$ . So $f$ is not onto $κ$ ; $κ$ stays a cardinal. The same cover bounds cofinalities, so cofinality is preserved. $□$

Proposition 4 (nice-name upper bound on $2^{ℵ_{0}}$ ). If $P$ is ccc with $∣ P ∣ = μ$ in $M$ , then $M [G] ⊨ 2^{ℵ_{0}} \leq (μ^{ℵ_{0}})^{M}$ .

Proof. Every $x \subseteq ω$ in $M [G]$ has a nice name $μ_{x} = ⋃_{n} {\overset{n}{ˇ}} \times A_{n}$ with each $A_{n}$ an antichain of $P$ (by the Truth Lemma, taking $A_{n}$ a maximal antichain of conditions forcing $\overset{n}{ˇ} \in μ_{x}$ ). By ccc each $A_{n}$ is countable, so a nice name is an $ω$ -sequence of countable subsets of $P$ . The number of countable subsets of $P$ is $μ^{ℵ_{0}}$ , and the number of $ω$ -sequences of them is $(μ^{ℵ_{0}})^{ℵ_{0}} = μ^{ℵ_{0}}$ . Distinct reals have distinct nice names, so $(2^{ℵ_{0}})^{M [G]} \leq (μ^{ℵ_{0}})^{M}$ , the right side computed in $M$ and preserved as a cardinal by Proposition 3. $□$

Proposition 5 (the Cohen model: $2^{ℵ_{0}} = ℵ_{2}$ ). If $M ⊨ ZFC + GCH$ and $G$ is generic for $P = Fn (ℵ_{2}^{M} \times ω, 2)$ , then $M [G] ⊨ 2^{ℵ_{0}} = ℵ_{2}$ .

Proof. Lower bound: the reals $c_{ξ}$ ( $ξ < ℵ_{2}$ ) are pairwise distinct (Exercise 3's density), and cardinals are preserved by Propositions 2-3, so $M [G] ⊨ 2^{ℵ_{0}} \geq ℵ_{2}$ . Upper bound: $∣ P ∣ = ∣ [ℵ_{2} \times ω]^{< ω} ∣ \cdot 2 = ℵ_{2}$ in $M$ , and GCH gives $ℵ_{2}^{ℵ_{0}} = ℵ_{2}$ (since $cf (ℵ_{2}) > ℵ_{0}$ and GCH), so Proposition 4 gives $M [G] ⊨ 2^{ℵ_{0}} \leq ℵ_{2}$ . The two bounds coincide. $□$

Proposition 6 (countably-closed forcing adds no reals). If $P$ is $< ω_{1}$ -closed in $M$ , then $(2^{ω})^{M [G]} = (2^{ω})^{M}$ .

Proof. Let $\overset{c}{˙}$ name a function $\overset{ω}{ˇ} \to \overset{ˇ}{2}$ and $p \in P$ . Recursively pick $p = p_{0} \geq p_{1} \geq \dots$ with $p_{n + 1}$ deciding $\overset{c}{˙} (\overset{n}{ˇ})$ (density of decision). By $< ω_{1}$ -closure the descending $ω$ -sequence has a lower bound $q \leq p_{n}$ for all $n$ . Then $q$ decides every $\overset{c}{˙} (\overset{n}{ˇ})$ , defining $g \in M$ with $g (n)$ the decided value, and $q ⊩ \overset{c}{˙} = \overset{g}{ˇ}$ . Thus densely many conditions force $\overset{c}{˙}$ to equal a ground-model real, so by the Truth Lemma every real of $M [G]$ lies in $M$ ; the converse inclusion is automatic. $□$

Connections Master

Forcing I: posets, generics, names, and the Fundamental Theorem 42.03.07 is the direct prerequisite supplying every tool used here. The generic extension $M [G]$ , the interpretation of names, and the Truth Lemma are imported wholesale; this unit adds only the chain-condition and counting layer on top. The nice-name machinery that proved Power Set in $M [G]$ there is the very same that bounds $2^{ℵ_{0}}$ here, and the closure dial that preserved small sequences there is what forces CH here. Forcing II is forcing I specialised to one poset, $Add (ω, ℵ_{2})$ , run for one purpose.
The constructible universe $L$ and the consistency of GCH 42.03.06 supplies the second half of the independence theorem and the dual technique. $L ⊨ GCH$ gives $Con (ZFC + CH)$ , while this unit's Cohen model gives $Con (ZFC + \neg CH)$ ; together they bracket CH as undecidable. Condensation in $L$ caps the continuum at $ℵ_{1}$ from below; the genericity-driven nice-name count lifts it to $ℵ_{2}$ from above. Cofinality and cardinal exponentiation 42.03.04 supplies König's inequality $cf (2^{ℵ_{0}}) > ℵ_{0}$ that constrains which cardinals the Cohen poset can force the continuum to be, and is sharpened by Easton's theorem (Theorem 4) realising the continuum function on the regulars.
Martin's axiom and iterated forcing 42.03.09 is the co-produced continuation that iterates the single Cohen step of this unit. A finite-support ccc iteration of length $ℵ_{2}$ forces Martin's axiom plus $2^{ℵ_{0}} = ℵ_{2}$ , settling the Suslin and Borel-conjecture problems and separating the cardinal invariants of the continuum; the preservation of ccc through the iteration is exactly the $Δ$ -system / antichain control developed here, and the two-step iteration $P * \dot{Q}$ generalises the product $Add (ω, ℵ_{2})$ . The axiom of choice 42.03.05 enters through the descent of Choice to $M [G]$ and, via symmetric submodels of a Cohen extension, supplies models of $\neg AC$ .

Historical & philosophical context Master

Paul Cohen proved the independence of the continuum hypothesis in 1963, announcing it in two notes to the Proceedings of the National Academy of Sciences and giving the full account in the 1966 monograph Set Theory and the Continuum Hypothesis ^{[Cohen 1963]}. He adjoined $ℵ_{2}$ mutually generic subsets of $ω$ to a countable standard model of ZF by finite conditions, the finiteness keeping the forcing countably-chained so that no cardinal collapsed and the new reals stayed $ℵ_{2}$ -many. Together with Gödel's 1938 proof that $L ⊨ GCH$ , this settled the first problem on Hilbert's 1900 list — not by deciding CH but by showing ZFC cannot. Cohen received the Fields Medal in 1966, the only one ever awarded for mathematical logic.

The method was reorganised at once. Azriel Lévy and Robert Solovay showed in 1967 that adding a large cardinal above the Cohen forcing leaves $\neg$ CH intact, so no large-cardinal axiom of the standard hierarchy decides CH; Solovay's 1970 model derived from an inaccessible a world where every set of reals is Lebesgue measurable ^{[Jech Ch. 15]}. The $Δ$ -system lemma used here goes back to Erdős and Rado's 1960 sunflower theorem. Donald Martin and Solovay isolated Martin's axiom as the iteration making the continuum large while preserving ccc, and Richard Laver, Saharon Shelah, and others extended iterated forcing to the cardinal invariants of the continuum, organised by the Cichoń diagram of Tomek Bartoszyński and Haim Judah. Whether CH has a determinate truth value beyond ZFC remains contested: Hugh Woodin's work on $Ω$ -logic and the $P_{m a x}$ axiom argues for $2^{ℵ_{0}} = ℵ_{2}$ as the privileged answer, while the set-theoretic multiverse of Joel Hamkins treats CH as a parameter varying across models, each Cohen and collapse extension a legitimate universe.

Bibliography Master

@book{kunen2011set,
  author    = {Kunen, Kenneth},
  title     = {Set Theory},
  series    = {Studies in Logic},
  volume    = {34},
  publisher = {College Publications},
  year      = {2011}
}

@book{jech2003set,
  author    = {Jech, Thomas},
  title     = {Set Theory},
  edition   = {3},
  series    = {Springer Monographs in Mathematics},
  publisher = {Springer},
  year      = {2003}
}

@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{levysolovay1967measurable,
  author  = {L\'evy, Azriel and Solovay, Robert M.},
  title   = {Measurable cardinals and the continuum hypothesis},
  journal = {Israel Journal of Mathematics},
  volume  = {5},
  year    = {1967},
  pages   = {234--248}
}

@article{erdosrado1960intersection,
  author  = {Erd\H{o}s, Paul and Rado, Richard},
  title   = {Intersection theorems for systems of sets},
  journal = {Journal of the London Mathematical Society},
  volume  = {35},
  year    = {1960},
  pages   = {85--90}
}

@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}
}

@book{bartoszynskijudah1995set,
  author    = {Bartoszy\'nski, Tomek and Judah, Haim},
  title     = {Set Theory: On the Structure of the Real Line},
  publisher = {A K Peters},
  year      = {1995}
}

Prerequisites

42.03.07

Tier anchors

beginner: Weaver 2014 *Forcing for Mathematicians* (World Scientific) Ch. 5-8 (adding Cohen reals by hand, the countable chain condition, why cardinals survive, and the failure of CH with no metatheory); Cohen 1966 *Set Theory and the Continuum Hypothesis* (Benjamin) Ch. IV (the original informal account of forcing many reals and counting them)
intermediate: Kunen 2011 *Set Theory* (College Publications) Ch. V §1-4 and Ch. VII (the Δ-system lemma, the ccc of Fn(I,2), cardinal and cofinality preservation by ccc forcing, the nice-name count bounding 2^{ℵ₀}, and the construction of a model of ¬CH by adding ℵ₂ Cohen reals); Jech 2003 *Set Theory* 3e (Springer) Ch. 15 (the Cohen model, ccc preservation, and Easton forcing on the regulars)
master: Kunen 2011 *Set Theory* (College Publications) Ch. V and VIII in full (the Δ-system lemma for the ccc of finite-partial-function posets, the cardinal-preservation theorem, the nice-name bound 2^{ℵ₀} ≤ |Fn(κ×ω,2)|^{ℵ₀} = κ^{ℵ₀}, the Cohen model for ¬CH, the countably-closed collapse forcing CH/GCH, and the independence of ◊ and the Suslin hypothesis); Jech 2003 *Set Theory* 3e (Springer) Ch. 15, 28-29 (the continuum function under forcing, Easton's theorem on the regulars, and cardinal invariants of the continuum); Bartoszyński-Judah 1995 *Set Theory: On the Structure of the Real Line* (A K Peters) for the Cichoń diagram and the independence of the cardinal invariants

References

Kunen, K. — Set Theory · College Publications, Studies in Logic 34 (2011, revised edition). Chapters V and VIII apply the forcing machinery of Chapter IV to the continuum problem. The countable chain condition (ccc): a forcing poset P is ccc if every antichain (set of pairwise-incompatible conditions) in P is countable. The key combinatorial input is the Δ-system lemma: every uncountable family of finite sets contains an uncountable subfamily forming a Δ-system (sunflower) — a family with a common pairwise-intersection root r such that the members minus r are pairwise disjoint. Applied to the domains of an uncountable antichain in Fn(I,2), the Δ-system lemma forces uncountably many conditions sharing a root yet pairwise incompatible, which on the finite root r ⊆ I leaves only finitely many functions r → 2, a contradiction; hence Fn(I,2) is ccc for every index set I. Cardinal preservation: a ccc poset preserves cardinals and cofinalities, because by the nice-name / countable-spread bound every function f : κ → λ of M[G] is covered by a function F : κ → [λ]^{≤ω} of M (each f(α) lies in a countable set F(α) computed from a maximal — hence countable — antichain deciding f(α)), so no cardinal is collapsed and no cofinality is changed. Adding Cohen reals: the poset Add(ω,κ) = Fn(κ×ω, 2) of finite partial functions κ×ω → 2 is ccc, and a generic G yields κ distinct Cohen reals c_ξ(n) = (⋃G)(ξ,n) for ξ < κ, pairwise distinct because the diagonal-difference sets are dense; hence M[G] ⊨ 2^{ℵ₀} ≥ κ. The nice-name count bounds the continuum from above: the number of nice names for subsets of ω is at most |Fn(κ×ω,2)|^{ℵ₀} = κ^{ℵ₀} computed in M, so under GCH in M with κ = ℵ₂ one gets M[G] ⊨ 2^{ℵ₀} = ℵ₂, and CH fails. Forcing CH and GCH: the countably-closed Levy collapse Fn(ω₁, 2^{ℵ₀}) (or Add(ω₁,1) over a suitable model) adds no reals and well-orders the old reals in order type ω₁, forcing CH; iterating on every regular forces GCH. Other independence results: ◊ and the Suslin hypothesis are independent (◊ holds in L and after the countably-closed collapse, SH holds after the ccc iteration forcing MA + ¬CH); Easton's theorem realizes any monotone class function on the regular cardinals respecting König's theorem as the continuum function. The book also develops the independence of the cardinal invariants of the continuum (the Cichoń diagram) by adding dominating, random, and Cohen reals.
Jech, T. — Set Theory · Springer Monographs in Mathematics, 3rd millennium edition (2003). Chapter 15 (Applications of Forcing) builds the Cohen model. The Cohen forcing for ¬CH adjoins ℵ₂ mutually generic subsets of ω to a ground model of GCH using the finite-support product Fn(ℵ₂ × ω, 2); the finite conditions make the poset ccc by the Δ-system lemma, so all cardinals are preserved, while the ℵ₂ reals witness 2^{ℵ₀} ≥ ℵ₂ and the nice-name count gives 2^{ℵ₀} ≤ (ℵ₂)^{ℵ₀} = ℵ₂, hence 2^{ℵ₀} = ℵ₂ exactly. The chapter proves the cardinal-and-cofinality preservation theorem for ccc forcing through the maximal-antichain / countable-cover argument, and contrasts it with <κ-closed forcing, which adds no new <κ-sequences and so forces CH or GCH by collapsing or coding without adding reals. Easton's theorem (Chapter 15) realizes the continuum function 2^{κ} on regular κ as any class function E with E(κ) ≤ E(λ) for κ ≤ λ and cf(E(κ)) > κ, by an Easton-support product of the Cohen posets Add(κ, E(κ)); this shows GCH can fail arbitrarily badly on the regulars while ZFC proves only monotonicity and König's inequality cf(2^κ) > κ. Chapters 28-29 develop cardinal invariants of the continuum — the bounding number 𝔟, dominating number 𝔡, the additivity and cofinality of measure and category — and their independence by iterated forcing, the Cichoń diagram organising the provable inequalities while forcing separates the rest. The combination of Cohen's ¬CH model with Gödel's L ⊨ GCH establishes the independence of CH from ZFC: Con(ZFC) implies both Con(ZFC + CH) and Con(ZFC + ¬CH).
Cohen, P. J. — The Independence of the Continuum Hypothesis · *Proceedings of the National Academy of Sciences* 50 (1963), 1143-1148, and 51 (1964), 105-110, with the full exposition in *Set Theory and the Continuum Hypothesis* (Benjamin, 1966). Cohen builds, from a countable standard model M of ZF, a generic extension in which the continuum hypothesis fails by adjoining ℵ₂^M mutually generic subsets of ω (Cohen reals) via finite conditions. A condition is a finite consistent set of forcing statements about the new sets, equivalently a finite partial function from ℵ₂ × ω to 2; the finite-condition structure is what keeps the forcing countably-chained, so that the cardinals ℵ₁ and ℵ₂ of M are not collapsed and the ℵ₂ new reals remain ℵ₂-many in the extension. Cohen's enumeration of conditions and his proof that a complete sequence (one meeting every requirement) yields a model in which the forced sentences are exactly the true ones is the original form of the truth lemma; the counting of the conditions and the names for reals bounds 2^{ℵ₀} above. The construction establishes Con(ZF) ⟹ Con(ZF + ¬CH); together with Gödel's L it gives the full independence of CH from ZFC. Cohen also indicated how to force CH and how the method settles the independence of the axiom of choice via symmetric submodels. For this work Cohen received the 1966 Fields Medal, the only one awarded for mathematical logic; the method became the universal instrument for independence results in set theory.

Estimated time

beginner: 20m
intermediate: 55m
master: 95m