42.03.09 · mathematical-logic / set-theory-forcing

Martin's Axiom, Iterated Forcing, and Large Cardinals

shipped3 tiersLean: none

Anchor (Master): Kunen 2011 *Set Theory* (College Publications) Ch. III, VIII, and X in full (Martin's axiom MA_κ and the cardinal invariant 𝔭=2^{ℵ₀} under MA, finite- and countable-support iteration, the Solovay-Tennenbaum theorem Con(MA+¬CH), and the large-cardinal hierarchy from inaccessible through measurable with the ultrapower characterization); Jech 2003 *Set Theory* 3e (Springer) Ch. 16-17, 20, and Part III (iterated and proper forcing, measurable cardinals and elementary embeddings j:V→M, the consistency-strength ladder, and determinacy); Shelah 1998 *Proper and Improper Forcing* 2e (Springer) Ch. I-III (proper forcing, the ℵ₁-preservation theorem, PFA); Kanamori 2009 *The Higher Infinite* 2e (Springer) Ch. 1-6, 32 (the large-cardinal hierarchy, measurable and supercompact cardinals, Woodin cardinals and the Martin-Steel theorem on projective determinacy)

Intuition Beginner

The previous units showed that the size of the real line is not decided by the usual rules: you can grow the universe by adding new reals and make the continuum hypothesis fail, or trim it and make it hold. That freedom is unsettling. If the rules leave the most basic question about infinity open, which extra rules should we adopt? This unit is about the two great answers set theory has developed, and how they organize an otherwise chaotic landscape.

The first answer is a new axiom that says, in effect, "stop forcing — the universe already contains every small generic object you could build." If a gentle construction of a certain controlled kind could produce some object meeting a modest list of demands, then that object is already here. This single principle, Martin's axiom, decides a long list of questions that the bare rules leave open, and it works by imagining many forcing steps stacked one after another rather than just one.

The second answer reaches upward instead of sideways. It posits infinities so vast that their mere existence cannot be proved from below, each one stronger than the last. These large infinities form a ruler. Any new assumption you might add can be measured against the ruler by asking which large infinity it is as risky as. The remarkable empirical fact is that almost everything lines up neatly along this single scale, turning a wilderness of independent statements into an ordered tower.

Visual Beginner

Two pictures organize the whole subject. On the left, Martin's axiom: a tall stack of gentle forcing steps, each adding a little, with the axiom saying the finished object at the top already sits in the universe. On the right, the ruler of large infinities: a vertical scale where each higher mark proves that the marks below it are safe, and every new assumption gets pinned to one height on the scale.

  MARTIN'S AXIOM                    THE LARGE-INFINITY RULER
  (iterate gently, sideways)        (climb upward in strength)

    step 1  add a little              supercompact   strongest
    step 2  add a little                   |
    step 3  add a little              Woodin
      .                                    |
      .     ... many steps ...        measurable
      .                                    |
   limit:  the object is             weakly compact
           ALREADY in the universe        |
                                      Mahlo
   "the universe is already               |
    generic for small gentle         inaccessible    weakest above ZFC
    constructions"                        ===
                                       plain ZFC

The table contrasts the two programs. They answer different needs: one settles questions about the real line at the level of the continuum; the other measures how strong an assumption is and reaches consequences far above the reals.

feature	Martin's axiom (sideways)	large infinities (upward)
what it adds	a genericity rule for small gentle forcings	enormous new infinities
built from	stacking many forcing steps	positing a top, then descending
main job	settle continuum-level questions	measure consistency strength
relation to the rules	consistent with the rules, like the rules plus a setting	each cannot be proved from the rules below

The key idea: forcing revealed the rules do not pin down the universe, and the response is a pair of programs — Martin's axiom organizing the real line sideways, large infinities organizing strength upward — that together hunt for the right extra rules.

Worked example Beginner

Let us see Martin's axiom do one concrete job: rule out a strange object called a gap. Picture two growing towers of sets of whole numbers. Tower $A$ climbs upward, each set containing the previous one; tower $B$ descends, each set inside the previous one; and every $A$ -set sits inside every $B$ -set with room to spare. The question: is there a single set $C$ slotted neatly between the two towers, above all of $A$ and below all of $B$ ? Sometimes the towers are short and a $C$ is plain; the worry is towers as long as the next size of infinity, where a $C$ might fail to exist — a gap.

Step 1. Set up a gentle search for $C$ . A promise is a finite snapshot: finitely many numbers declared "in $C$ " and finitely many declared "out", consistent with sitting between the towers so far.

Step 2. For each level of tower $A$ , the demand " $C$ contains this whole $A$ -level" can always be met by extending a finite promise; likewise " $C$ avoids enough of this $B$ -level". Each demand is a reachable region.

Step 3. The number of demands is one per tower level — fewer than the continuum if the towers are shorter than the continuum.

Step 4. The gentle search is of the controlled kind Martin's axiom governs, and there are fewer demands than the continuum, so the axiom hands us a finished filter meeting them all. Reading off that filter gives the set $C$ .

What this tells us: under Martin's axiom plus the failure of the continuum hypothesis, no gap of length below the continuum exists — a question the plain rules leave open is settled by the single principle "small gentle searches already succeed".

Check your understanding Beginner

Exercise (easy, multiple-choice).

Why is stacking many gentle forcing steps the engine behind Martin's axiom?

Because a single step can already meet every demand at once.
Because each step can be aimed at one demand, and a long stack can be aimed, one step at a time, at every small forcing and every demand.
Because stacking steps deletes the reals.
Because gentle steps are forbidden by the rules.

Hint

A bookkeeping list lets the stack anticipate every gentle forcing along the way.

Answer

B. A long stack of gentle steps, guided by a bookkeeping list, can devote individual steps to each small forcing and each demand, so the finished universe is generic for all of them at once — which is what Martin's axiom asserts. Feedback-correct: iteration is how one builds a universe satisfying the axiom. Feedback-wrong: no single step handles everything (A is false), and stacking neither deletes reals nor is forbidden (C, D are false).

Formal definition Intermediate+

Work in ZFC with the forcing apparatus of 42.03.07 and the ccc machinery of 42.03.08: posets $(P, \leq, 1_{P})$ , compatibility, antichains, dense sets, $M$ -generic filters, $P$ -names, the generic extension $M [G]$ , and the countable chain condition (every antichain is countable).

For a cardinal $κ$ , Martin's axiom at $κ$ , written $MA_{κ}$ , is the statement: for every ccc poset $P$ and every family $D$ of at most $κ$ dense subsets of $P$ , there is a filter $G \subseteq P$ such that $G \cap D \neq = \emptyset$ for all $D \in D$ . Such a $G$ is called $D$ -generic. Martin's axiom is $MA = MA_{< 2^{ℵ_{0}}}$ , that is, $MA_{κ}$ holds for every $κ < 2^{ℵ_{0}}$ . By the Rasiowa-Sikorski lemma $MA_{ℵ_{0}}$ is a theorem of ZFC, and $MA_{2^{ℵ_{0}}}$ is false (it fails already for $P = Add (ω, 1)$ ), so $MA$ is informative precisely when $2^{ℵ_{0}} > ℵ_{1}$ ; under CH, $MA$ holds vacuously.

A two-step iteration $P * \dot{Q}$ has as conditions pairs $(p, \overset{q}{˙})$ with $p \in P$ and $\overset{q}{˙}$ a $P$ -name forced by $p$ to be a condition of $\dot{Q}$ , ordered by $(p^{'}, \overset{q}{˙}^{'}) \leq (p, \overset{q}{˙})$ iff $p^{'} \leq p$ and $p^{'} ⊩ \overset{q}{˙}^{'} \leq \overset{q}{˙}$ . A forcing iteration of length $α$ with support type $s$ is a sequence $(P_{ξ} : ξ \leq α)$ with $P_{ξ + 1} = P_{ξ} * \dot{Q}_{ξ}$ and, at limits $η$ , $P_{η}$ the inverse limit restricted to conditions whose support lies in the chosen ideal — finite support ( $∣ supp (p) ∣ < ℵ_{0}$ ) or countable support ( $∣ supp (p) ∣ \leq ℵ_{0}$ ).

A cardinal $κ$ is inaccessible if it is uncountable, regular, and strong-limit ( $λ < κ \Rightarrow 2^{λ} < κ$ ); measurable if it carries a $κ$ -complete non-principal ultrafilter $U$ on $κ$ . The critical point of a non-identity elementary embedding $j : V \to M$ into a transitive class $M$ is the least ordinal moved.

Key theorem with proof Intermediate+

The organising fact is that finite-support iteration keeps the chain condition, which is what lets a single iteration force Martin's axiom. The chain condition survives because supports are finite and antichains are controlled by the $Δ$ -system lemma of 42.03.08, exactly as for the one-step Cohen poset.

Theorem (finite-support ccc iteration is ccc). Let $(P_{ξ} : ξ \leq α)$ be a finite-support iteration such that for every $ξ < α$ one has $⊩_{P_{ξ}} \dot{Q}_{ξ}$ is ccc. Then $P_{α}$ is ccc. ^{[Kunen Ch. VIII]}

Proof. By induction on $α$ . Successor steps reduce to the two-step case: if $P_{ξ}$ is ccc and $⊩_{P_{ξ}} \dot{Q}_{ξ}$ is ccc, then $P_{ξ} * \dot{Q}_{ξ}$ is ccc, because an antichain of $P_{ξ} * \dot{Q}_{ξ}$ of size $ℵ_{1}$ projects (after refining the second coordinates with a name for a maximal antichain of $\dot{Q}_{ξ}$ , countable by the assumed ccc) to an uncountable antichain of $P_{ξ}$ or to an uncountable antichain of $\dot{Q}_{ξ}$ in some extension, each excluded. For the limit $α$ , suppose $A = {(p_{β}) : β < ω_{1}}$ is an antichain of $P_{α}$ . Each $p_{β}$ has finite support $s_{β} \subseteq α$ . Apply the $Δ$ -system lemma to ${s_{β} : β < ω_{1}}$ : an uncountable $W$ with ${s_{β} : β \in W}$ a $Δ$ -system of root $r$ , where $r \subseteq α$ is finite. The root $r$ is contained in some $P_{γ}$ with $γ < α$ (as $cf (α)$ may be uncountable, finiteness of $r$ places it below $α$ ). Restricting the conditions to $r$ gives ${p_{β} ↾ r : β \in W}$ , $ℵ_{1}$ -many conditions of the ccc poset $P_{γ}$ (induction hypothesis), so two of them, $p_{β} ↾ r$ and $p_{β^{'}} ↾ r$ , are compatible in $P_{γ}$ via some $q$ . Off the root the supports $s_{β}, s_{β^{'}}$ are disjoint, so $q$ together with the disjoint parts amalgamates to a common extension of $p_{β}$ and $p_{β^{'}}$ in $P_{α}$ . That contradicts antichain-hood; hence $∣ A ∣ \leq ℵ_{0}$ . $□$

Bridge. The finite-support theorem is the foundational reason a single forcing can install Martin's axiom: the chain condition that 42.03.08 secured for one Cohen step now survives stacking $ℵ_{2}$ steps, so a bookkeeping iteration can meet every ccc forcing and every dense family without ever collapsing $ℵ_{1}$ . This is exactly the mechanism that builds toward the Solovay-Tennenbaum theorem in the Advanced results — once $P_{ℵ_{2}}$ is ccc and anticipates all small ccc data, $M [G_{ℵ_{2}}] ⊨ MA + 2^{ℵ_{0}} = ℵ_{2}$ . The construction generalises the single Cohen poset of 42.03.08 from one step to a transfinite stack: $Add (ω, ℵ_{2})$ is the special case where every $\dot{Q}_{ξ}$ is Cohen forcing and nothing is anticipated. The central insight is that genericity is iterable — a long enough controlled iteration makes the universe generic for an entire class of forcings at once, which is precisely what Martin's axiom asserts about the ground model after the fact. This appears again in the countable-support iteration of proper forcings that yields PFA, where the preservation theorem upgrades from ccc to properness and the supports from finite to countable, and the same amalgamation-over-a-root argument is replaced by the elementary-submodel genericity of Shelah's properness. Putting these together, the modern theory of the continuum is the theory of which forcings can be iterated while preserving $ℵ_{1}$ , and Martin's axiom is the first and gentlest such iteration program.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Prove the Rasiowa-Sikorski lemma, i.e. $MA_{ℵ_{0}}$ : for any poset $P$ and countably many dense sets $D_{0}, D_{1}, \dots$ , there is a filter meeting every $D_{n}$ . (No ccc needed.)

Hint

Build a descending sequence of conditions, one entering each $D_{n}$ in turn.

Answer

Start with any $p_{0} \in P$ . Given $p_{n}$ , since $D_{n}$ is dense there is $p_{n + 1} \leq p_{n}$ with $p_{n + 1} \in D_{n}$ . The sequence $p_{0} \geq p_{1} \geq \dots$ is a descending chain; let $G = {q \in P : \exists n (p_{n} \leq q)}$ , the upward closure of the chain. Then $G$ is a filter: it is upward closed by construction, and any two $p_{m}, p_{n}$ have the common lower bound $p_{m a x (m, n)} \in G$ , so any two elements of $G$ are compatible within $G$ . And $p_{n + 1} \in G \cap D_{n}$ for each $n$ , so $G$ meets every $D_{n}$ . Rubric: full credit for the descending construction, the verification that the upward closure is a filter, and the meeting of each $D_{n}$ .

Exercise 4 (medium, symbolic).

Show that $MA_{ℵ_{1}}$ implies the product of two ccc posets is ccc. (Hence under $MA + \neg CH$ , ccc is productive.)

Hint

If $P \times Q$ had an uncountable antichain, use $MA_{ℵ_{1}}$ on $P$ to find a filter meeting $ℵ_{1}$ dense sets coding the antichain's first coordinates, and derive an uncountable antichain in $Q$ .

Answer

Suppose $P, Q$ are ccc but $P \times Q$ has an antichain ${(p_{ξ}, q_{ξ}) : ξ < ω_{1}}$ . The sets $E_{ξ} = {p \in P : p \leq p_{ξ}$ or $p ⊥ p_{ξ}}$ are dense in $P$ . By $MA_{ℵ_{1}}$ applied to the ccc poset $P$ and the $ℵ_{1}$ dense sets $E_{ξ}$ , there is a filter $G$ meeting each $E_{ξ}$ ; let $S = {ξ : G ∋$ some $p \leq p_{ξ}}$ , so $G$ "selects" the $p_{ξ}$ with $ξ \in S$ and these are pairwise compatible (members of a filter), hence ${q_{ξ} : ξ \in S}$ must be pairwise incompatible in $Q$ (else some $(p_{ξ}, q_{ξ}), (p_{η}, q_{η})$ would be compatible in the product). If $S$ were uncountable this is an uncountable antichain in the ccc $Q$ , impossible; if $S$ is countable, the complementary $ℵ_{1}$ -many $ξ \in / S$ have $G$ meeting $E_{ξ}$ through a $p ⊥ p_{ξ}$ , and a symmetric count via a generic filter on $Q$ forces an uncountable antichain in $P$ . Either way a ccc factor has an uncountable antichain, a contradiction; so $P \times Q$ is ccc. Rubric: credit for the dense sets $E_{ξ}$ , the $MA_{ℵ_{1}}$ filter, and the transfer of the uncountable antichain into one ccc factor.

Exercise 5 (medium, symbolic).

Let $j : V \to M$ be a nontrivial elementary embedding into a transitive class $M$ with critical point $κ$ . Show $κ$ is inaccessible in $V$ : regular and strong-limit.

Hint

Use elementarity and $V_{κ} \subseteq M$ with $V_{κ} = (V_{κ})^{M}$ ; transfer regularity and the strong-limit property between $V$ and $M$ via $j$ .

Answer

Since $κ = crit (j)$ , $j$ fixes every $α < κ$ and $j (κ) > κ$ . Regular: if $κ = sup_{i < λ} β_{i}$ with $λ < κ$ and each $β_{i} < κ$ , then $j (β_{i}) = β_{i}$ and $j (λ) = λ$ , so $j (κ) = j (sup β_{i}) = sup_{i < λ} j (β_{i}) = sup_{i < λ} β_{i} = κ$ , contradicting $j (κ) > κ$ ; hence $κ$ has no short cofinal sequence and is regular. Strong-limit: for $λ < κ$ , $j (2^{λ}) = 2^{j (λ)} = 2^{λ}$ by elementarity and $j ↾ λ = id$ , and since $j$ fixes ordinals below $κ$ but moves $κ$ , the cardinal $2^{λ}$ cannot equal or exceed $κ$ (else it would be moved or pin $κ$ below the critical point); so $2^{λ} < κ$ . Thus $κ$ is regular and strong-limit, i.e. inaccessible (it is uncountable since the embedding is nontrivial). Rubric: credit for using $crit (j) = κ$ to fix small ordinals, the cofinality argument for regularity, and the elementarity argument for strong-limit.

Exercise 6 (hard, symbolic).

Sketch why a measurable cardinal $κ$ gives a nontrivial elementary embedding $j : V \to M$ with $crit (j) = κ$ , via the ultrapower by a $κ$ -complete non-principal ultrafilter $U$ .

Hint

Form $Ult (V, U)$ with Łoś's theorem; $κ$ -completeness gives well-foundedness; take the transitive collapse and let $j$ be the canonical embedding $[c_{x}] \mapsto$ the constant function.

Answer

Form the ultrapower $Ult (V, U)$ of functions $f : κ \to V$ modulo $=_{U}$ (agreement on a set in $U$ ), with membership $[f] \in_{U} [g]$ iff ${α : f (α) \in g (α)} \in U$ . Łoś's theorem gives $Ult (V, U) ⊨ φ ([f_{1}], \dots)$ iff ${α : φ (f_{1} (α), \dots)} \in U$ , so the diagonal map $j (x) = [c_{x}]$ (constant function $c_{x}$ ) is elementary: $φ (x)$ holds in $V$ iff ${α : φ (x)} = κ \in U$ iff $Ult ⊨ φ (j (x))$ . $κ$ -completeness of $U$ (closure under intersections of $< κ$ members) makes $\in_{U}$ well-founded — any descending $\in_{U}$ -sequence would, by $κ$ -completeness, induce a genuinely descending $\in$ -sequence on a set in $U$ , impossible — so $Ult$ is isomorphic to a transitive class $M$ ; let $j : V \to M$ be the composite. The critical point is $κ$ : each $α < κ$ is fixed because $U$ is $κ$ -complete and non-principal (the function $id$ witnesses $j (α) = α$ for $α < κ$ but $[id]$ is a new ordinal with $κ \leq [id] < j (κ)$ , so $j (κ) > κ$ ). Hence $j$ is nontrivial with $crit (j) = κ$ . Rubric: credit for Łoś giving elementarity, $κ$ -completeness giving well-foundedness and transitive collapse, and the identity function witnessing $crit (j) = κ$ .

Exercise 7 (hard, symbolic).

Explain why no large-cardinal axiom of the standard hierarchy decides CH, by sketching the Levy-Solovay theorem: small forcing preserves measurability.

Hint

If $∣ P ∣ < κ$ and $κ$ is measurable via $U$ , lift $U$ to a $κ$ -complete ultrafilter in $V [G]$ ; then force CH or $\neg$ CH by Cohen-type forcing of size $< κ$ .

Answer

Let $κ$ be measurable via a $κ$ -complete non-principal ultrafilter $U$ , and let $P$ be a forcing with $∣ P ∣ < κ$ ("small"). In $V [G]$ define $\overset{ˉ}{U} = {X \subseteq κ : \exists Y \in U, Y \subseteq X}$ , the upward closure of $U$ . Because $∣ P ∣ < κ$ and $κ$ is regular, every new subset of $κ$ in $V [G]$ is decided by an antichain of size $< κ$ , so $κ$ -completeness of $U$ transfers: $\overset{ˉ}{U}$ is a $κ$ -complete non-principal ultrafilter on $κ$ in $V [G]$ , witnessing that $κ$ stays measurable. Now take $P$ to be a Cohen forcing $Add (ω, ℵ_{2})$ of size $ℵ_{2} < κ$ (forcing $\neg$ CH) or a countably-closed collapse of size $< κ$ (forcing CH); both are small relative to $κ$ , so $κ$ remains measurable in $V [G]$ while CH takes either truth value. Hence measurability is compatible with CH and with $\neg$ CH, so a measurable cardinal cannot decide CH; the same small-forcing argument applies to every large cardinal whose defining embedding/ultrafilter is $κ$ -complete, i.e. the entire standard hierarchy. Rubric: credit for the lifted ultrafilter $\overset{ˉ}{U}$ , the small-forcing transfer of $κ$ -completeness, and the conclusion that CH is forceable either way above the large cardinal.

Advanced results Master

The independence of CH from 42.03.08 left set theory with a question of method: which extra axioms should organize the universe? Two answers, forcing axioms and large cardinals, structure everything below, and they meet in the theory of consistency strength and determinacy.

Theorem 1 (Solovay-Tennenbaum; $Con (ZFC) \Rightarrow Con (ZFC + MA + \neg CH)$ ). Over $M ⊨ ZFC + GCH$ , perform a finite-support iteration $(P_{ξ} : ξ \leq ℵ_{2})$ of length $ℵ_{2}$ in which a bookkeeping function $h : ℵ_{2} \to ℵ_{2} \times ℵ_{2}$ guarantees that every ccc forcing $\dot{Q}$ of hereditary size $< ℵ_{2}$ appearing in any intermediate model, together with every family of $< ℵ_{2}$ dense sets, is enumerated and forced at some stage $\dot{Q}_{ξ}$ ^{[Kunen Ch. VIII]}. Each $\dot{Q}_{ξ}$ is ccc, so by the Key theorem $P_{ℵ_{2}}$ is ccc; cardinals are preserved (the ccc preservation of 42.03.08). A nice-name count gives $2^{ℵ_{0}} = ℵ_{2}$ in $M [G]$ , and the bookkeeping ensures every ccc poset of size $< 2^{ℵ_{0}}$ with $< 2^{ℵ_{0}}$ dense sets was met along the way, with the relevant filter surviving into the final model. Hence $M [G] ⊨ MA + 2^{ℵ_{0}} = ℵ_{2}$ , so $MA + \neg CH$ is consistent relative to ZFC.

Theorem 2 (consequences of MA + ¬CH). Under $MA + \neg CH$ : the union of fewer than $2^{ℵ_{0}}$ Lebesgue-null sets is null and of fewer than $2^{ℵ_{0}}$ meagre sets is meagre, so the additivity of measure and of category both equal $2^{ℵ_{0}}$ ^{[Kunen Ch. III]}; every ccc poset has Knaster's property K (every uncountable subset has an uncountable centered subset), whence ccc is productive; there are no Suslin lines, so the Suslin hypothesis holds; and $p = 2^{ℵ_{0}}$ , collapsing the small cardinal invariants of the continuum to the continuum and trivialising the lower part of the Cichoń diagram of 42.03.08. The single principle "the universe is already generic for small ccc forcings" thus settles a long list of questions the bare axioms leave open, each by exhibiting a ccc forcing whose generic produces the desired object and invoking $MA$ to place that object in the ground universe.

Theorem 3 (proper forcing and PFA). Shelah's class of proper forcings — those preserving stationary subsets of $[λ]^{ℵ_{0}}$ , equivalently those for which club-many countable $N ≺ H_{θ}$ admit $(N, P)$ -generic conditions — is closed under countable-support iteration and preserves $ℵ_{1}$ ^{[Jech Ch. 16-17]}. The Proper Forcing Axiom PFA is $MA$ -style genericity for all proper posets and $ℵ_{1}$ -many dense sets; it implies $2^{ℵ_{0}} = ℵ_{2}$ , the Suslin hypothesis, that all $ℵ_{1}$ -Aronszajn trees are special, and Baumgartner's theorem that all $ℵ_{1}$ -dense sets of reals are order-isomorphic. PFA's consistency is obtained from a supercompact cardinal, the first appearance of the upper hierarchy controlling a statement about the reals; it sharpens $MA$ by replacing ccc (finite-support) with proper (countable-support), at the cost of large-cardinal strength.

Theorem 4 (the large-cardinal hierarchy and consistency strength). The cardinals inaccessible $<$ Mahlo $<$ weakly compact $<$ Ramsey $<$ measurable $<$ Woodin $<$ supercompact form a strictly increasing scale of consistency strength: each proves $Con$ of the theory asserting any smaller one exists ^{[Kanamori Ch. 1-6]}. A measurable cardinal is characterised by a nontrivial elementary embedding $j : V \to M = Ult (V, U)$ with $crit (j) = κ$ (Exercise 6), and by Scott's theorem a measurable refutes $V = L$ ; stronger cardinals demand $M$ to be closed under longer sequences ( $κ$ supercompact iff for all $λ$ there is $j$ with $crit κ$ , $j (κ) > λ$ , $M^{λ} \subseteq M$ ). The central empirical fact: every known natural extension of ZFC is equiconsistent with some level of this hierarchy, so large cardinals serve as the canonical yardstick of consistency strength, and the apparent linearity of that yardstick is one of the deepest unexplained regularities in foundations.

Theorem 5 (large cardinals and determinacy). The Axiom of Determinacy AD contradicts the axiom of choice 42.03.05 but holds in $L (R)$ under sufficient large cardinals ^{[Kanamori Ch. 32]}. The Martin-Steel theorem: $n$ Woodin cardinals with a measurable above yield $Π_{n + 1}^{1}$ -determinacy, so infinitely many Woodin cardinals give full projective determinacy PD, whence every projective set of reals is Lebesgue measurable, has the Baire property, and has the perfect-set property — the regularity properties that $V = L$ falsifies and that ZFC alone cannot settle. Solovay's model, built from a single inaccessible, makes every set of reals Lebesgue measurable (dropping choice). These results realise the program of large cardinals as new axioms: by reaching upward in consistency strength one settles downward questions about the definable reals that forcing showed ZFC cannot decide.

Synthesis. The foundational reason set theory did not end with the independence of CH is that forcing is not only a destabiliser but a constructor: the same finite-support ccc iteration that 42.03.08 used for one Cohen step, iterated $ℵ_{2}$ times under bookkeeping, builds a universe satisfying Martin's axiom, and the central insight is that genericity is iterable — a long enough chain-condition-preserving iteration makes the ground universe generic for an entire class of forcings, which is exactly what a forcing axiom asserts after the fact. This is exactly the seam between the two great programs: $MA$ , PFA, and the proper-forcing axioms organize the universe sideways at the level of the continuum, fixing $2^{ℵ_{0}} = ℵ_{2}$ and the cardinal invariants, while the large-cardinal hierarchy organizes it upward by consistency strength, and putting these together the two meet precisely at PFA, whose consistency needs a supercompact, and at determinacy, where Woodin cardinals settle the regularity of the projective reals.

The large-cardinal route is dual to the forcing route of 42.03.08: forcing changes the model to alter a statement's truth, an elementary embedding $j : V \to M$ holds the universe fixed and reflects truth between $V$ and an inner model, so the constructible universe of 42.03.06 caps the continuum from below while a measurable refutes $V = L$ from above, and the bridge is consistency strength — Levy-Solovay shows small forcing is absorbed, so large cardinals cannot decide CH, and the modern search (Woodin's $Ω$ -logic, the $P_{m a x}$ axiom, the multiverse) is the attempt to find the right axioms that the cofinality and cardinal-arithmetic constraints of 42.03.04 leave open. This generalises the single forcing step of 42.03.08 into set theory's permanent program: to organize the universe by the two perpendicular rulers of forcing-iterability and consistency strength, and to ask which extra axioms those rulers recommend.

Full proof set Master

Proposition 1 (two-step ccc iteration is ccc). If $P$ is ccc and $⊩_{P} \dot{Q}$ is ccc, then $P * \dot{Q}$ is ccc.

Proof. Suppose ${(p_{ξ}, \overset{q}{˙}_{ξ}) : ξ < ω_{1}}$ is an antichain of $P * \dot{Q}$ . For each pair $ξ \neq = η$ , $(p_{ξ}, \overset{q}{˙}_{ξ}) ⊥ (p_{η}, \overset{q}{˙}_{η})$ means no $r \leq p_{ξ}, p_{η}$ forces $\overset{q}{˙}_{ξ}, \overset{q}{˙}_{η}$ compatible. Work in $V^{P}$ : let $G$ be $P$ -generic and set $S = {ξ : p_{ξ} \in G}$ . The names $\overset{q}{˙}_{ξ}$ for $ξ \in S$ interpret to conditions $q_{ξ}^{G}$ of $\dot{Q} [G]$ , pairwise incompatible (if $q_{ξ}^{G}, q_{η}^{G}$ had a common extension, a condition in $G$ below $p_{ξ}, p_{η}$ would force compatibility). Since $\dot{Q}$ is ccc in $V [G]$ , ${q_{ξ}^{G} : ξ \in S}$ is countable, so $S$ is countable in $V [G]$ for every generic $G$ . Thus $⊩_{P} ∣ {ξ : p_{ξ} \in \dot{G}} ∣ \leq ℵ_{0}$ . But $P$ is ccc, so by the maximal-antichain/countable-cover bound of 42.03.08 the set ${ξ : p_{ξ} \in \dot{G}}$ ranges, across generics, over a ground-model-countable union; were ${p_{ξ} : ξ < ω_{1}}$ to have $ℵ_{1}$ -many distinct conditions each in some generic, an uncountable antichain of $P$ would result, contradicting ccc unless uncountably many $p_{ξ}$ coincide — and equal $p_{ξ}$ force their $\overset{q}{˙}_{ξ}$ pairwise incompatible, an uncountable antichain of the ccc $\dot{Q}$ . Either way a contradiction; so $∣ {(p_{ξ}, \overset{q}{˙}_{ξ})} ∣ \leq ℵ_{0}$ . $□$

Proposition 2 ( $MA_{ℵ_{0}}$ is a theorem of ZFC). For any poset $P$ and countably many dense $D_{n} \subseteq P$ , a filter meets every $D_{n}$ .

Proof. Build $p_{0} \geq p_{1} \geq \dots$ with $p_{n + 1} \in D_{n}$ by density (Exercise 3). The upward closure $G$ of the chain is a filter and $p_{n + 1} \in G \cap D_{n}$ . $□$

Proposition 3 ( $MA_{ℵ_{1}}$ implies the Suslin hypothesis). Under $MA_{ℵ_{1}}$ there is no Suslin line; equivalently every ccc dense complete linear order without endpoints is separable.

Proof. A Suslin line yields a Suslin tree $T$ : a tree of height $ω_{1}$ with no uncountable chain or antichain. View $T$ as a forcing poset (ordered by reverse tree-order, so extensions are stronger). $T$ is ccc precisely because it has no uncountable antichain. For each $α < ω_{1}$ the set $D_{α} = {t \in T : ht (t) > α}$ is dense (every node has extensions at arbitrarily high levels, as $T$ has height $ω_{1}$ and no maximal nodes once pruned). By $MA_{ℵ_{1}}$ applied to the ccc poset $T$ and the $ℵ_{1}$ dense sets ${D_{α}}$ , there is a filter $G$ meeting each $D_{α}$ . But a filter in a tree is a chain (any two elements are compatible, hence tree-comparable), and meeting every $D_{α}$ makes it cofinal in the $ω_{1}$ levels, so $G$ is an uncountable chain in $T$ — contradicting Suslinity. Hence no Suslin tree, and no Suslin line, exists. $□$

Proposition 4 (Scott: a measurable refutes $V = L$ ). If there is a measurable cardinal, then $V \neq = L$ .

Proof. Let $κ$ be measurable via $U$ and $j : V \to M = Ult (V, U)$ with $crit (j) = κ$ (Exercise 6). Suppose $V = L$ . Then $M \subseteq L = V$ , and since $M$ is a transitive class model of ZFC containing all ordinals, $M = L = V$ as well (the constructible universe is the minimal such; $L^{M} = L$ by absoluteness of the $L$ -hierarchy 42.03.06). Elementarity gives $M ⊨ V = L$ , so $j : L \to L$ is a nontrivial elementary embedding of $L$ into itself with critical point $κ$ . By Kunen's argument (or directly: the least ordinal moved cannot exist for an embedding of $L$ definable from $U$ , as $j (κ)$ would have to be the $κ$ -th element of a definable well-order it also fixes below $κ$ , forcing $j (κ) = κ$ ), no such embedding exists. The contradiction shows $V \neq = L$ . $□$

Proposition 5 (inaccessibility of a critical point). If $j : V \to M$ is elementary, nontrivial, $M$ transitive, with $crit (j) = κ$ , then $κ$ is inaccessible.

Proof. Regularity and the strong-limit property follow from elementarity and the fixing of ordinals below $κ$ , as in Exercise 5: a cofinal map $λ \to κ$ with $λ < κ$ would be fixed by $j$ and force $j (κ) = κ$ ; and $j (2^{λ}) = 2^{λ}$ for $λ < κ$ keeps $2^{λ}$ below the critical point. Uncountability is immediate as $κ > ω$ (every natural number is fixed). $□$

Connections Master

Forcing II: Cohen and the independence of CH 42.03.08 is the direct prerequisite and the seed of every construction here. The ccc preservation and $Δ$ -system control proved there for the single Cohen poset are exactly what the finite-support iteration theorem of this unit inherits; the Solovay-Tennenbaum model is $Add (ω, ℵ_{2})$ generalised from one step to an $ℵ_{2}$ -length bookkeeping iteration, and the cardinal invariants of the Cichoń diagram that 42.03.08 separated by single forcings are collapsed to the continuum by Martin's axiom here. Forcing axioms are the program that answers the question 42.03.08 raised — which axioms settle what ZFC leaves open at the level of the reals.
The forcing machinery 42.03.07 supplies the two-step iteration $P * \dot{Q}$ and the name calculus that the iteration theorem rests on; the generic extension, names, and the Fundamental Theorem are imported wholesale, and the iteration $(P_{ξ})$ is built by repeating the single-step $M [G]$ construction transfinitely. The constructible universe and GCH 42.03.06 is the dual technique: $L$ caps the continuum and supports $◊$ and a Suslin tree, while a measurable cardinal refutes $V = L$ (Proposition 4), so the large-cardinal hierarchy reaches strictly past the constructible universe that forcing axioms and large cardinals both transcend.
Cofinality and cardinal exponentiation 42.03.04 supplies the regularity and König constraints that bound what the continuum can be even under $MA$ and large cardinals, and Easton's theorem there is the precise statement of how free the continuum function on the regulars remains; the modern search for the right axioms (Woodin's $Ω$ -logic) operates within exactly those constraints. The axiom of choice 42.03.05 enters twice: determinacy AD contradicts choice yet holds in $L (R)$ under Woodin cardinals, and the ultrapower construction of the embedding $j : V \to M$ for a measurable cardinal uses choice essentially. The model-theoretic stability landscape (the trichotomy and Shelah's classification of first-order theories by counting types) is the parallel program in 42.02.05 pending and 42.02.08 pending: there too a single dividing line — stability, or o-minimal tameness — organizes an entire field by how wildly its objects can proliferate, and Shelah's hand is on both the proper-forcing and the classification-theory rulers.

Historical & philosophical context Master

Donald Martin and Robert Solovay isolated Martin's axiom in their 1970 paper on internal Cohen extensions, where the Solovay-Tennenbaum theorem that a finite-support ccc iteration forces $MA + \neg CH$ first appeared ^{[Kunen Ch. VIII]}. The axiom crystallised the idea that one could legislate the universe to be generic for small forcings, settling the Suslin hypothesis — open since Mikhail Suslin's 1920 problem — in the affirmative under $MA + \neg CH$ . Saharon Shelah's theory of proper forcing, developed through the 1970s and presented in Proper and Improper Forcing, extended iteration from ccc to the much wider class preserving $ℵ_{1}$ , and the Proper Forcing Axiom of James Baumgartner gave the strongest of the standard forcing axioms, its consistency drawn from a supercompact cardinal.

The large-cardinal hierarchy grew in parallel. Stanisław Ulam introduced measurable cardinals in 1930 via measure-theoretic problems; Dana Scott proved in 1961 that a measurable refutes $V = L$ , the first theorem showing large cardinals constrain the constructible universe, using the ultrapower embedding $j : V \to M$ that became the organising tool of the field ^{[Kanamori Ch. 1-6]}. The consistency-strength ladder and its near-linearity were charted by Azriel Lévy, Robert Solovay, Jack Silver, and others; Lévy and Solovay's 1967 theorem that small forcing preserves measurability showed large cardinals cannot decide CH. Donald Martin, John Steel, and Hugh Woodin proved in 1985 that Woodin cardinals yield projective determinacy, the high point of the program connecting the upper hierarchy to the regularity of the definable reals. Whether these axioms fix a determinate truth value for CH remains contested: Woodin's $Ω$ -logic and $P_{m a x}$ argue for $2^{ℵ_{0}} = ℵ_{2}$ , while Joel Hamkins's set-theoretic multiverse treats CH as a parameter varying across equally legitimate universes.

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{martinsolovay1970internal,
  author  = {Martin, Donald A. and Solovay, Robert M.},
  title   = {Internal Cohen extensions},
  journal = {Annals of Mathematical Logic},
  volume  = {2},
  year    = {1970},
  pages   = {143--178}
}

@book{shelah1998proper,
  author    = {Shelah, Saharon},
  title     = {Proper and Improper Forcing},
  edition   = {2},
  series    = {Perspectives in Mathematical Logic},
  publisher = {Springer},
  year      = {1998}
}

@book{kanamori2009higher,
  author    = {Kanamori, Akihiro},
  title     = {The Higher Infinite},
  edition   = {2},
  series    = {Springer Monographs in Mathematics},
  publisher = {Springer},
  year      = {2009}
}

@article{scott1961measurable,
  author  = {Scott, Dana},
  title   = {Measurable cardinals and constructible sets},
  journal = {Bulletin de l'Acad\'emie Polonaise des Sciences},
  volume  = {9},
  year    = {1961},
  pages   = {521--524}
}

@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{martinsteel1989projective,
  author  = {Martin, Donald A. and Steel, John R.},
  title   = {A proof of projective determinacy},
  journal = {Journal of the American Mathematical Society},
  volume  = {2},
  year    = {1989},
  pages   = {71--125}
}

@book{woodin1999axiom,
  author    = {Woodin, W. Hugh},
  title     = {The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal},
  series    = {De Gruyter Series in Logic and Its Applications},
  publisher = {Walter de Gruyter},
  year      = {1999}
}

Prerequisites

42.03.08

Tier anchors

beginner: Weaver 2014 *Forcing for Mathematicians* (World Scientific) Ch. 9-12 (Martin's axiom as 'the generic already lives in the universe', the ccc product lemma, and the Suslin line ruled out by hand, with no metatheory); Cohen 1966 *Set Theory and the Continuum Hypothesis* (Benjamin) Ch. IV (the original informal account that one forcing step can be iterated)
intermediate: Kunen 2011 *Set Theory* (College Publications) Ch. III §3 (Martin's axiom and its consequences: the union of <2^{ℵ₀} meagre sets is meagre, ccc products, no Suslin lines) and Ch. V §4-6 and Ch. VIII (finite-support iteration of ccc forcing, the Solovay-Tennenbaum consistency of MA+¬CH, two-step iteration P*Q̇); Jech 2003 *Set Theory* 3e (Springer) Ch. 16-17 (iterated forcing, Martin's axiom, and the basics of proper forcing)
master: Kunen 2011 *Set Theory* (College Publications) Ch. III, VIII, and X in full (Martin's axiom MA_κ and the cardinal invariant 𝔭=2^{ℵ₀} under MA, finite- and countable-support iteration, the Solovay-Tennenbaum theorem Con(MA+¬CH), and the large-cardinal hierarchy from inaccessible through measurable with the ultrapower characterization); Jech 2003 *Set Theory* 3e (Springer) Ch. 16-17, 20, and Part III (iterated and proper forcing, measurable cardinals and elementary embeddings j:V→M, the consistency-strength ladder, and determinacy); Shelah 1998 *Proper and Improper Forcing* 2e (Springer) Ch. I-III (proper forcing, the ℵ₁-preservation theorem, PFA); Kanamori 2009 *The Higher Infinite* 2e (Springer) Ch. 1-6, 32 (the large-cardinal hierarchy, measurable and supercompact cardinals, Woodin cardinals and the Martin-Steel theorem on projective determinacy)

References

Kunen, K. — Set Theory · College Publications, Studies in Logic 34 (2011, revised edition). Chapter III develops Martin's axiom. MA_κ asserts: for every ccc partial order P and every family D of at most κ dense subsets of P there is a filter G ⊆ P meeting every member of D; Martin's axiom MA is MA_κ for all κ < 2^{ℵ₀}, equivalently MA_{<2^{ℵ₀}}. MA_{ℵ₀} is a theorem of ZFC (the Rasiowa-Sikorski lemma), and MA + CH is trivially true, so the interesting hypothesis is MA + ¬CH. Reading MA as 'the universe is already generic for small ccc forcings': any configuration that a ccc forcing of size <2^{ℵ₀} could create, meeting <2^{ℵ₀} dense requirements, already exists in V. Consequences of MA+¬CH proved in Ch. III: the union of fewer than 2^{ℵ₀} Lebesgue-null sets is null and the union of fewer than 2^{ℵ₀} meagre sets is meagre (so 2^{ℵ₀} is regular and the additivity of measure and category equals 2^{ℵ₀}); every ccc partial order has property K and the product of ccc partial orders is ccc (Knaster's property); there are no Suslin lines, i.e. the Suslin hypothesis holds; all ℵ₁-dense sets of reals are order-isomorphic (Baumgartner, by proper forcing in the PFA strengthening); and the pseudo-intersection number 𝔭 equals 2^{ℵ₀}, collapsing a swathe of the cardinal invariants of the continuum to the continuum. Chapter V §4-6 and Chapter VIII develop iterated forcing: the finite-support iteration of ccc forcings is ccc (the key lemma, via the Δ-system argument applied to the supports), and the two-step iteration P * Q̇ where Q̇ is a P-name for a forcing. The Solovay-Tennenbaum theorem: iterating all ccc forcings of size <ℵ₂ with finite support over a model of GCH, using a bookkeeping function to anticipate every ccc forcing and every dense family, yields a model of MA + 2^{ℵ₀} = ℵ₂; hence Con(ZFC) ⟹ Con(ZFC + MA + ¬CH). Chapter VIII treats countable-support iteration and the requirement of proper forcing to preserve ℵ₁ through an ω₂-length iteration. Chapter X develops the large-cardinal hierarchy: inaccessible (regular strong-limit) cardinals and the consistency strength Con(ZFC) of a single inaccessible; Mahlo cardinals (the inaccessibles below are stationary); weakly compact cardinals (the tree property, or κ → (κ)^2_2); measurable cardinals κ carrying a κ-complete non-principal ultrafilter U, equivalently admitting a nontrivial elementary embedding j : V → M with critical point κ into a transitive class M (the ultrapower Ult(V,U) by Łoś's theorem); and the ascending strength of Ramsey, Woodin, and supercompact cardinals. Each large cardinal proves the consistency of the smaller ones (the consistency-strength ladder), and the empirically observed near-linear ordering of all known natural consistency strengths along this ladder is one of the central phenomena of the subject.
Jech, T. — Set Theory · Springer Monographs in Mathematics, 3rd millennium edition (2003). Chapters 16-17 build iterated forcing and Martin's axiom; Chapter 20 and Part III develop measurable cardinals, elementary embeddings, and determinacy. The two-step iteration P * Q̇: conditions are pairs (p, q̇) with p ∈ P and q̇ a P-name forced to be a condition of Q̇; a generic for P * Q̇ decomposes as G * H where G is P-generic and H is Q̇[G]-generic over V[G], the iteration theorem. A forcing iteration of length α is a sequence (P_ξ : ξ ≤ α) with P_{ξ+1} = P_ξ * Q̇_ξ and limits taken with a chosen support (finite, countable, or Easton). Finite-support iteration of ccc forcings is ccc; this drives the Solovay-Tennenbaum construction of a model of MA + ¬CH. For longer (ω₂-length, ℵ₁-preserving) iterations of non-ccc forcings one needs Shelah's proper forcing: P is proper if it preserves stationary subsets of [λ]^{ℵ₀}, equivalently for club-many countable elementary submodels N ≺ H_θ every condition has an (N,P)-generic extension; proper forcings are closed under countable-support iteration and preserve ℵ₁. The Proper Forcing Axiom PFA asserts MA-style genericity for all proper forcings and ℵ₁-many dense sets; PFA implies 2^{ℵ₀} = ℵ₂, the Suslin hypothesis, that all ℵ₁-Aronszajn trees are special, and Baumgartner's theorem that all ℵ₁-dense real order types coincide; the consistency of PFA is obtained from a supercompact cardinal. Chapter 20 and Part III: a measurable cardinal yields a nontrivial elementary embedding j : V → M = Ult(V,U) with crit(j) = κ; the embedding reflects large-cardinal and combinatorial properties and gives 0# and the failure of V = L above a measurable. Determinacy: the Axiom of Determinacy AD contradicts choice but holds in L(ℝ) under large cardinals; the Martin-Steel theorem (1985, with Woodin) derives projective determinacy PD from infinitely many Woodin cardinals, and AD^{L(ℝ)} from ω Woodins with a measurable above; Solovay's model from an inaccessible makes every set of reals Lebesgue measurable, with the Baire property and the perfect set property. Large cardinals settle questions L cannot (projective regularity, the size of the continuum's invariants) and are organized by consistency strength.
Kanamori, A. — The Higher Infinite · Springer Monographs in Mathematics, 2nd edition (2009). The definitive treatment of large cardinals and their elementary-embedding characterizations. An inaccessible cardinal κ is regular and strong-limit (2^λ < κ for λ < κ); V_κ is then a model of ZFC, so Con(ZFC) does not prove the existence of an inaccessible (second incompleteness). A Mahlo cardinal is inaccessible with the inaccessibles below it stationary. A weakly compact cardinal is inaccessible with the tree property — every κ-tree has a cofinal branch — equivalently κ → (κ)^2_2, equivalently Π^1_1-indescribable. A measurable cardinal κ carries a κ-complete non-principal ultrafilter; by Łoś and Scott the ultrapower Ult(V,U) is well-founded, its transitive collapse M gives a nontrivial elementary j : V → M with critical point κ, and conversely any such embedding yields a measurable; Scott's theorem (1961) that a measurable refutes V = L was the first demonstration that large cardinals constrain the constructible universe. Ramsey cardinals (κ → (κ)^{<ω}_2) sit between weakly compact and measurable in consistency strength. Strong, Woodin, and supercompact cardinals are defined by the existence of elementary embeddings j : V → M with M closed under longer sequences and higher agreement with V: κ is supercompact if for every λ there is j : V → M with crit κ, j(κ) > λ, and M^λ ⊆ M. The consistency-strength ladder: each level proves Con of the levels below, and (the empirical central fact of the subject) every known natural extension of ZFC by 'large' hypotheses is equiconsistent with some level of this hierarchy, so the large cardinals provide a measuring stick — a near-linear scale of consistency strength — against which independent statements are calibrated. Chapter 32 and the determinacy chapters give the Martin-Steel-Woodin theorems: n Woodin cardinals with a measurable above yield Π^1_{n+1}-determinacy, so infinitely many Woodins give full projective determinacy, and the structural consequences (every projective set is Lebesgue measurable, has the Baire and perfect-set properties), realizing the program of large cardinals as the new axioms settling the regularity questions of descriptive set theory that ZFC and V = L leave open or decide wrongly.

Estimated time

beginner: 22m
intermediate: 55m
master: 100m