42.03.06 · mathematical-logic / set-theory-forcing

The Constructible Universe L and the Consistency of GCH

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Anchor (Master): Kunen 2011 *Set Theory* (College Publications) Ch. II §1-6 and the exercises on relative constructibility and the fine structure pointers, with Ch. IV for the absoluteness preliminaries; Jech 2003 *Set Theory* 3e (Springer) Ch. 13 (constructibility, the condensation lemma, GCH and ◊ in L), Ch. 18 (the covering theorem and 0#), and Ch. 27 (◊ and □); Devlin 1984 *Constructibility* (Springer Perspectives in Logic) for the fine-structure development

Intuition Beginner

The previous units built the universe of sets one layer at a time, throwing in every subset of each layer as you climb. Gödel asked a sharper question: what if, at each new layer, you only admit the subsets you can actually describe using the sets already built? You forbid the wild, nameless subsets and keep only the ones a sentence can pin down. The collection you end up with is called the constructible universe, written $L$ , and it is the most economical world of sets that still obeys all the usual rules.

Think of it as building a library where every book must be written, page by page, from books already on the shelves — no book may simply appear by fiat. Because every set in $L$ comes with a recipe, you can line all of them up in a single master order: first the sets describable at the bottom, then the next batch, and so on. That master order is exactly what lets you always "pick one from each box," so the awkward selection problem of the previous unit solves itself for free inside $L$ .

The payoff is enormous. Inside this disciplined library, the question "how many subsets does an infinite set have?" — which looked wide open — gets a clean answer: the smallest one the rules permit. Gödel used this to show that assuming the cleanest answer can never lead to a contradiction. The next units flip the construction over to show the opposite answer is also safe, and the two halves together reveal that the question has no forced answer at all.

Visual Beginner

A set lands in $L$ only when some sentence, using sets from earlier layers, picks it out exactly. The picture contrasts the full universe (keep all subsets) with the constructible one (keep only the describable subsets).

   FULL UNIVERSE  V                CONSTRUCTIBLE  L
   (keep EVERY subset)             (keep only DESCRIBABLE subsets)

   level 3:  all subsets           level 3:  only the named subsets
   level 2:  all subsets   <----   level 2:  only the named subsets   (slimmer)
   level 1:  all subsets           level 1:  only the named subsets
   level 0:  empty                 level 0:  empty

   every nameless subset survives  a subset survives only with a recipe

The table records how the two libraries differ as you climb. At the very bottom they agree; the gap opens once a layer has subsets no sentence can single out.

layer	full universe keeps	constructible universe keeps
level 0	nothing (start empty)	nothing (start empty)
level 1	the one possible subset	the one possible subset
a finite layer	every subset	every subset (all are describable)
an infinite layer	every subset, named or not	only the subsets some sentence defines

The key idea: $L$ is the universe you get by admitting a set only when you can write down a description that selects it from the layer below.

Worked example Beginner

Let us build the first few layers of the constructible library by hand and watch where the "describable only" rule starts to bite. Write a describable subset of a layer as one that some sentence, using members of that layer, picks out exactly.

Step 1. Start at the bottom with the empty layer: there is nothing in it yet. The only subset of nothing is the empty collection itself, and the sentence "no member at all" describes it. So the next layer holds exactly one thing: the empty set. Call it $0$ .

Step 2. Now the layer holds the single item $0$ . Its subsets are "the empty one" and "the one containing $0$ ." Both are describable — "contains nothing" and "contains $0$ " — so the next layer holds two items.

Step 3. Continue. The layer now holds two items. Every subset of a two-item layer can be named by listing which of the two it contains, so all four subsets are describable, and they all enter the next layer. So far, at every finite stage, nothing is lost: each subset has a short description.

Step 4. The difference only appears once a layer is infinite. An infinite layer has so many subsets that no finite stock of sentences can name them all; the nameless ones are left out of $L$ . The full universe would keep them; the constructible library does not.

Step 5. Because each item that does enter carries a description, you can order all of them: sort by the layer where they first appear, and within a layer by their describing sentence. That single ordering reaches every constructible set.

What this tells us: at finite layers $L$ and the full universe look identical, but at infinite layers $L$ keeps only the nameable subsets — and that very discipline hands you a master ordering of everything inside.

Check your understanding Beginner

Exercise (easy, multiple-choice).

Why does the constructible universe come with a single master ordering of all its sets?

Because it contains only finitely many sets.
Because each set carries a description and a layer where it first appears, which can be sorted.
Because all its sets are numbers.
Because the full universe has no sets.

Hint

Think about what extra information every constructible set carries that an arbitrary set does not.

Answer

B. Each constructible set has a stage where it first appears and a describing sentence, so sorting by stage and then by description lines them all up. Feedback-correct: this built-in ordering is exactly what makes selection automatic inside $L$ . Feedback-wrong: the constructible universe is infinite (A is false) and holds far more than numbers (C is false).

Formal definition Intermediate+

All of the following takes place inside ZF; ordinals, transfinite recursion, and rank are as in 42.03.02, cardinals and cofinality as in 42.03.03 and 42.03.04, and the Axiom of Choice is not assumed for the ambient theory — its status inside $L$ is one of the results. The only primitive is $\in$ .

For a set $X$ , the definable power set is $$ \operatorname{Def}(X) = \big{, a \subseteq X : a = {, y \in X : (X, \in) \models \varphi[y, p_1, \dots, p_n] ,} \text{ for some formula } \varphi \text{ and } p_1, \dots, p_n \in X ,\big}. $$ The satisfaction relation $(X, \in) ⊨ φ [\overset{a}{ˉ}]$ for set structures is itself a definable relation (Tarski's inductive truth definition for a set model, cross-ref 42.01.06 pending), so $Def$ is a definable class function. The constructible hierarchy is defined by transfinite recursion: $$ L_0 = \varnothing, \qquad L_{\alpha+1} = \operatorname{Def}(L_\alpha), \qquad L_\lambda = \bigcup_{\alpha < \lambda} L_\alpha \ \ (\lambda \text{ limit}), $$ and the constructible universe is the class $L = ⋃_{α \in On} L_{α}$ . Each $L_{α}$ is transitive, $α \subseteq L_{α}$ , $L_{α} \cap On = α$ , and $α \leq β \Rightarrow L_{α} \subseteq L_{β}$ ; for $α \geq ω$ one has $∣ L_{α} ∣ = ∣ α ∣$ , the discipline of keeping only definable subsets capping the size of each level.

The relativisation $φ^{M}$ of a formula $φ$ to a class $M$ restricts every quantifier to $M$ ( $\forall x ψ$ becomes $\forall x (x \in M \to ψ^{M})$ ). A formula is $Δ_{0}$ (or $Σ_{0}$ ) if all its quantifiers are bounded, $\forall x \in y$ and $\exists x \in y$ ; the Lévy hierarchy then sets $Σ_{n + 1}$ to be formulas $\exists \overset{x}{ˉ} ψ$ with $ψ \in Π_{n}$ , and $Π_{n + 1}$ dually. A formula $φ (\overset{x}{ˉ})$ is absolute between transitive classes $M \subseteq N$ when $φ^{M} (\overset{a}{ˉ}) \leftrightarrow φ^{N} (\overset{a}{ˉ})$ for all $\overset{a}{ˉ} \in M$ . The basic absoluteness facts (cross-ref [42.01]) are: every $Δ_{0}$ formula is absolute between transitive models; $Σ_{1}$ formulas are upward absolute and $Π_{1}$ formulas downward absolute. The relations " $x$ is an ordinal," " $x$ is transitive," " $z = {x, y}$ ," " $z = ⋃ x$ ," " $x$ is a function," and " $y = ω$ " are $Δ_{0}$ (or $Δ_{1}$ ) hence absolute; the relations " $y = P (x)$ ," " $κ$ is a cardinal," and " $x$ is constructible" are not absolute between arbitrary transitive models.

An inner model is a transitive class $M$ with $On \subseteq M$ that satisfies every axiom of ZF (i.e. $φ^{M}$ holds for each ZF axiom $φ$ ). The axiom of constructibility is the sentence $V = L$ : every set is constructible, $\forall x \exists α (x \in L_{α})$ . The canonical well-ordering $<_{L}$ of $L$ is defined by recursion on levels: order $L_{α + 1} ∖ L_{α}$ by the least Gödel number of a defining formula together with the (already-defined) ordering of parameter tuples from $L_{α}$ , and rank earlier levels first. This $<_{L}$ is a definable (parameter-free) well-ordering of the entire class $L$ .

Counterexamples to common slips Intermediate+

" $Def (X) = P (X)$ for infinite $X$ ." False, and this is the engine of the theory. For infinite $X$ there are $2^{∣ X ∣}$ subsets but only $∣ X ∣$ formulas-with-parameters, so $∣ Def (X) ∣ = ∣ X ∣ < ∣ P (X) ∣$ . The full power set is not definable; $Def$ keeps strictly fewer subsets.
" $V = L$ is just an abbreviation we can always assume." $V = L$ is a substantive axiom, independent of ZFC. It holds in $L$ but is refuted by, e.g., a measurable cardinal. Assuming it is a genuine restriction that decides CH, $♢$ , and $□$ , not a definitional convenience.
"Cardinals are absolute, so $ℵ_{1}^{L} = ℵ_{1}$ ." Cardinality is not absolute downward in general: an ordinal $L$ thinks is uncountable can become countable in $V$ . One always has $ℵ_{1}^{L} \leq ℵ_{1}$ , with equality failing exactly when $ω_{1}$ is collapsed (as forcing or $0^{#}$ can arrange). Absoluteness of cardinals holds only upward for the "is a cardinal" predicate being $Π_{1}$ .
" $L$ depends on the ambient $V$ you build it in." The hierarchy $L_{α}$ is defined by an absolute recursion, so $(L)^{L} = L$ and $(L_{α})^{M} = L_{α}$ for every inner model $M$ . $L$ is the same class computed in any inner model — this absoluteness is precisely what makes $L ⊨ V = L$ and what makes $L$ the least inner model.

Key theorem with proof Intermediate+

The organising fact is the condensation lemma: it converts the metamathematical move "take an elementary submodel" into the internal statement that the submodel is a level of the hierarchy, and from that single conversion both $V = L$ inside $L$ and the GCH bound fall out.

Theorem (the Condensation Lemma). Let $α$ be a limit ordinal and $M ≺ (L_{α}, \in)$ an elementary substructure. Then the Mostowski transitive collapse of $M$ is $(L_{β}, \in)$ for some $β \leq α$ , and the collapse map is the identity on $M \cap On$ restricted to its transitive part ^{[Kunen Ch. II]}.

Proof. Since $\in$ is set-like, well-founded, and extensional on $M$ (extensionality holds in $L_{α}$ and is inherited by the elementary $M$ ), the Mostowski collapse theorem (cross-ref 42.03.01) gives a unique transitive set $N$ and an isomorphism $π : (M, \in) \to (N, \in)$ . It remains to identify $N$ as some $L_{β}$ . The key input is that the predicate " $y = L_{ξ}$ " and the satisfaction relation for the structures $(L_{ξ}, \in)$ are $Δ_{1}$ over any inner model: the defining recursion for $L$ uses only the absolute operation $Def$ and the absolute recursion theorem, so " $v$ is the $ξ$ -th constructible level" is expressed by a $Σ_{1}$ formula with a $Π_{1}$ equivalent. Consequently the sentence "every set lies in some $L_{ξ}$ ," i.e. $V = L$ relativised, is true in $(L_{α}, \in)$ (each element of $L_{α}$ appears at some level $< α$ , and $L_{α}$ verifies this by the absoluteness of the level function). By elementarity $M$ satisfies the same sentence, and since $π$ is an isomorphism, the transitive $N$ satisfies $V = L$ as well: every element of $N$ is $L_{ξ}$ -constructible for some $ξ \in N \cap On$ .

Let $β = N \cap On = sup (M \cap On collapsed)$ , an ordinal because $N$ is transitive. The $Δ_{1}$ absoluteness of the level function means $π$ commutes with it: for $ξ \in N \cap On$ , the object $N$ believes to be $L_{ξ}$ is the genuine $L_{ξ}$ (a $Δ_{1}$ relation is absolute between the transitive $N$ and $V$ ). Hence $N = {x : x \in L_{ξ} for some ξ < β} = L_{β}$ . Finally $β \leq α$ because $M \subseteq L_{α}$ forces every ordinal of $N$ below $α$ , the collapse not increasing ordinals. $□$

Bridge. Condensation is the foundational reason the constructible universe is rigid enough to decide its own cardinal arithmetic: an elementary submodel of a level is forced to be a level, so no constructible subset of $ω_{α}$ can hide above its natural stage. This is exactly the mechanism that builds toward $L ⊨ GCH$ in the Advanced results — a constructible $A \subseteq ω_{α}$ is captured in a Skolem hull of size $ℵ_{α}$ , which condensation collapses to some $L_{β}$ with $β < ω_{α + 1}$ , so $A$ appears by stage $ω_{α + 1}$ and the count of such $A$ is $ℵ_{α + 1}$ . The lemma generalises the Löwenheim-Skolem method of [42.01] from arbitrary structures to the internal levels of $L$ , and it is dual to the well-ordering construction: where $<_{L}$ orders $L$ by the order of generation, condensation reads that generation backwards, recognising any elementary fragment as an initial generation. The central insight is that absoluteness of the defining recursion makes "be an elementary submodel" and "be a level $L_{β}$ " the same predicate on transitive sets satisfying $V = L$ . This appears again in the proof that $L$ is the least inner model and in Jensen's derivation of $♢$ and $□$ , where condensation is the sole combinatorial input. Putting these together, every structural property of $L$ — its global well-order, its GCH, its fine-structural principles — is a corollary of one lemma about collapsing elementary submodels.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Prove that $L_{α} \cap On = α$ for every ordinal $α$ ; conclude $On \subseteq L$ .

Hint

Induct on $α$ . At a successor, show $α \in Def (L_{α})$ and that no new ordinal beyond $α$ is added.

Answer

Induct on $α$ . Base: $L_{0} \cap On = \emptyset = 0$ . Limit $λ$ : $L_{λ} \cap On = ⋃_{α < λ} (L_{α} \cap On) = ⋃_{α < λ} α = λ$ . Successor: assume $L_{α} \cap On = α$ . Every ordinal $γ < α$ lies in $L_{α} \subseteq L_{α + 1}$ . The ordinal $α$ itself is the set ${γ \in L_{α} : γ is an ordinal}$ , definable over $L_{α}$ by the $Δ_{0}$ formula "is an ordinal," so $α \in Def (L_{α}) = L_{α + 1}$ . No ordinal $δ > α$ enters: a member of $L_{α + 1}$ is a subset of $L_{α}$ , and an ordinal $δ \subseteq L_{α}$ has all its elements $< α$ ordinals, forcing $δ \leq α$ . Hence $L_{α + 1} \cap On = α + 1$ . Taking the union over all $α$ gives $On \subseteq L$ . Rubric: full credit for the three induction cases, the definability of $α$ over $L_{α}$ , and the bound $δ \leq α$ .

Exercise 4 (medium, symbolic).

Show that $Def (X)$ contains every finite subset of $X$ and contains $X$ itself, and that for infinite $X$ , $∣ Def (X) ∣ = ∣ X ∣$ .

Hint

A finite set is defined by an explicit disjunction of equalities to parameters; count formulas-with-parameters for the cardinality.

Answer

For $a = {p_{1}, \dots, p_{k}} \subseteq X$ , the formula $φ (y) \equiv (y = p_{1} \lor \dots \lor y = p_{k})$ with parameters $p_{i} \in X$ defines $a$ over $(X, \in)$ , so $a \in Def (X)$ ; the formula $y = y$ defines $X$ . For cardinality: a member of $Def (X)$ is specified by a formula $φ$ (countably many, drawn from a fixed countable language) together with a finite parameter tuple from $X$ . The set of pairs (formula, finite tuple from $X$ ) has size $ℵ_{0} \cdot ∣ X^{< ω} ∣ = ℵ_{0} \cdot ∣ X ∣ = ∣ X ∣$ for infinite $X$ (using $∣ X^{< ω} ∣ = ∣ X ∣$ from 42.03.03), and the map (formula, tuple) $\mapsto$ defined set is a surjection onto $Def (X)$ , so $∣ Def (X) ∣ \leq ∣ X ∣$ ; the reverse $∣ X ∣ \leq ∣ Def (X) ∣$ holds since each singleton ${p} \in Def (X)$ . Hence $∣ Def (X) ∣ = ∣ X ∣$ . Rubric: full credit for the finite-set and whole-set definitions and the formula-times-parameters count.

Exercise 5 (medium, symbolic).

Prove that $<_{L}$ well-orders $L$ and that its restriction to each $L_{α}$ is a set well-ordering, definable uniformly in $α$ .

Hint

Order first by the least level of appearance, then lexicographically by (Gödel number of formula, $<_{L}$ -order of parameter tuple).

Answer

Define $<_{L}$ by recursion on levels. Suppose $<_{L} ↾ L_{α}$ is already a well-ordering. Each $x \in L_{α + 1} ∖ L_{α}$ is ${y \in L_{α} : (L_{α}, \in) ⊨ φ [y, \overset{p}{ˉ}]}$ for some pairs $(φ, \overset{p}{ˉ})$ with $\overset{p}{ˉ} \in L_{α}$ ; assign to $x$ the $<$ -least such pair under the ordering: compare Gödel number of $φ$ first, then the parameter tuple $\overset{p}{ˉ}$ lexicographically under $<_{L} ↾ L_{α}$ . Order $L_{α + 1}$ by: $x <_{L} x^{'}$ iff $x$ appears at an earlier level, or both first appear at level $α + 1$ and $x$ 's assigned least pair precedes $x^{'}$ 's. This is a linear order, and it is a well-order: any nonempty $S \subseteq L_{α + 1}$ has a least level of appearance among its members, and within that level the assigned pairs are well-ordered (Gödel numbers in $ω$ , tuples by the inductive well-order). At limits take the union of the level-orders, consistent because each extends the previous. The whole $<_{L}$ is a well-ordering of $L$ : any nonempty constructible $S$ has members of least level $β$ , and $<_{L} ↾ L_{β}$ picks the least. Definability uniformly in $α$ holds because $Def$ , satisfaction, and Gödel numbering are all absolute and arithmetical. Rubric: full credit for the level-then-pair ordering, the well-order verification, and the uniform definability remark.

Exercise 6 (hard, symbolic).

Prove that $L$ is the least inner model: if $M$ is any transitive class with $On \subseteq M$ and $M ⊨ ZF$ , then $L \subseteq M$ .

Hint

Show by induction on $α$ that $L_{α} \in M$ and $(L_{α})^{M} = L_{α}$ , using that the defining recursion for $L$ is absolute for inner models.

Answer

The operation $Def$ is absolute for transitive models of (enough of) ZF: $Def (X)$ is defined from the satisfaction relation for the set structure $(X, \in)$ , itself a $Δ_{1}$ notion built by an absolute recursion on formulas, so $Def (X)^{M} = Def (X)$ whenever $X \in M$ and $M$ is a transitive model. The level recursion $L_{0} = \emptyset$ , $L_{α + 1} = Def (L_{α})$ , $L_{λ} = ⋃_{α < λ} L_{α}$ is then computed identically inside $M$ . Induct on $α$ : $L_{0} = \emptyset \in M$ . Successor: if $L_{α} \in M$ then $Def (L_{α})^{M} = Def (L_{α}) = L_{α + 1}$ , and $M ⊨ ZF$ provides this definable power set as a member, so $L_{α + 1} \in M$ . Limit: $On \subseteq M$ gives $λ \in M$ , and Replacement in $M$ collects ${L_{α} : α < λ}$ (the level function is definable in $M$ ), so $L_{λ} = ⋃_{α < λ} L_{α} \in M$ . Since each $L_{α} \subseteq M$ (transitivity of $M$ plus $L_{α} \in M$ ) and $L = ⋃_{α} L_{α}$ with $On \subseteq M$ , every constructible set lies in $M$ . Hence $L \subseteq M$ . Rubric: full credit for the absoluteness of $Def$ /the level recursion, the three induction cases using ZF in $M$ , and the conclusion $L \subseteq M$ .

Exercise 7 (hard, symbolic).

Assume $V = L$ . Prove the GCH bound $2^{ℵ_{α}} = ℵ_{α + 1}$ for all $α$ , granting the condensation lemma: every $A \subseteq ω_{α}$ lies in some $L_{β}$ with $β < ω_{α + 1}$ .

Hint

Count the subsets of $ω_{α}$ that appear by stage $ω_{α + 1}$ using $∣ L_{ω_{α + 1}} ∣ = ℵ_{α + 1}$ , then add König's lower bound from 42.03.04.

Answer

Under $V = L$ every $A \subseteq ω_{α}$ is constructible, so $A \in L_{γ}$ for some $γ$ ; take a Skolem hull $M ≺ L_{γ}$ of size $ℵ_{α}$ containing $ω_{α} \cup {A}$ (possible since $∣ ω_{α} ∣ = ℵ_{α}$ ). By condensation $M$ collapses to some $L_{β}$ , and because the collapse fixes $ω_{α}$ pointwise (its elements are below the critical point) it fixes $A$ , so $A \in L_{β}$ with $∣ L_{β} ∣ = ∣ M ∣ = ℵ_{α}$ , whence $β < ω_{α + 1}$ (an ordinal of cardinality $ℵ_{α}$ is $< ω_{α + 1}$ ). Therefore every $A \subseteq ω_{α}$ lies in $L_{ω_{α + 1}}$ , giving $P (ω_{α}) \subseteq L_{ω_{α + 1}}$ and $2^{ℵ_{α}} = ∣ P (ω_{α}) ∣ \leq ∣ L_{ω_{α + 1}} ∣ = ∣ ω_{α + 1} ∣ = ℵ_{α + 1}$ . The reverse $2^{ℵ_{α}} \geq ℵ_{α + 1}$ is Cantor ( $2^{ℵ_{α}} > ℵ_{α}$ forces $\geq ℵ_{α + 1}$ ). Hence $2^{ℵ_{α}} = ℵ_{α + 1}$ . Rubric: full credit for the hull-and-condensation capture, the cardinality bound $β < ω_{α + 1}$ , the count $∣ L_{ω_{α + 1}} ∣ = ℵ_{α + 1}$ , and Cantor's lower bound.

Advanced results Master

The constructible universe is the canonical inner model: built by an absolute recursion, well-ordered by a definable order, and pinned down to a single GCH-respecting value of every exponential by condensation. The results below assemble Gödel's relative consistency theorem and then mark the boundary where $L$ ceases to see the universe.

Theorem 1 ( $L$ is an inner model of ZF). The class $L$ is transitive, contains every ordinal, and satisfies every axiom of ZF ^{[Kunen Ch. II]}. Extensionality and Foundation are inherited from $V$ by transitivity; Pairing, Union, and Infinity hold because the relevant witnesses are definable over some $L_{α}$ hence constructible; Separation and Replacement hold because the Lévy-reflection theorem provides, for any constructible parameters and formula, a level $L_{α}$ over which the instance reflects, and $Def (L_{α})$ supplies the separated or replaced set; Power Set holds because $P (x) \cap L$ , the constructible subsets of a constructible $x$ , all appear by some bounded level $L_{α}$ (a consequence of condensation bounding the stage of each constructible subset) and are collected by $Def$ at the next level. The verification is uniform and absolute, so $(L)^{L} = L$ and the relativised axioms hold.

Theorem 2 ( $L ⊨ V = L$ , via absoluteness). The sentence $V = L$ holds in $L$ ^{[Jech Ch. 13]}. The level function $α \mapsto L_{α}$ is defined by a $Σ_{1}$ recursion using only the absolute operation $Def$ , so $(L_{α})^{L} = L_{α}$ for every $α$ ; consequently the class $L$ computed inside $L$ is again the genuine $L$ , $(L)^{L} = L$ . Every $x \in L$ satisfies $x \in L_{α}$ for some $α$ , and this is witnessed inside $L$ , so $L ⊨ \forall x \exists α (x \in L_{α})$ , i.e. $L ⊨ V = L$ . This is the self-similarity that makes $L$ a minimal and rigid model: it knows that it is constructible.

Theorem 3 ( $L ⊨ AC$ ; the definable global well-ordering). The order $<_{L}$ is a definable well-ordering of $L$ without parameters, and $L ⊨$ " $<_{L}$ well-orders the universe" ^{[Gödel 1940]}. A definable well-ordering of the universe yields a choice function on every family (select the $<_{L}$ -least element of each member), so $L ⊨ AC$ — indeed $L$ satisfies the strong global choice principle "there is a definable well-ordering of $V$ ." Since $Con (ZF)$ gives $Con (ZF + V = L)$ and $V = L$ proves AC, this is Gödel's relative consistency of choice: $Con (ZF) \Rightarrow Con (ZFC)$ (cross-ref the converse arrow $Con (ZF) \Rightarrow Con (ZF + \neg AC)$ of 42.03.05).

Theorem 4 ( $L ⊨ GCH$ ; Gödel 1938). In $L$ , $2^{ℵ_{α}} = ℵ_{α + 1}$ for every $α$ ^{[Gödel 1940]}. By condensation a constructible $A \subseteq ω_{α}$ is captured in an elementary hull of cardinality $ℵ_{α}$ collapsing to some $L_{β}$ with $β < ω_{α + 1}$ , so $P (ω_{α})^{L} \subseteq L_{ω_{α + 1}}$ and $2^{ℵ_{α}} \leq ∣ L_{ω_{α + 1}} ∣ = ℵ_{α + 1}$ ; Cantor gives the reverse. Combining Theorems 3 and 4: $Con (ZF) \Rightarrow Con (ZFC + GCH)$ , the relative consistency of the generalised continuum hypothesis. Because GCH implies the singular cardinals hypothesis and collapses the gimel function $ℷ (κ) = κ^{+}$ of 42.03.04, $L$ is a model in which all of cardinal exponentiation reduces to the successor operation.

Theorem 5 ( $♢$ and $□$ hold in $L$ ; the limits of $L$ ). In $L$ the combinatorial principle $♢$ (a sequence $⟨ A_{δ} : δ < ω_{1} ⟩$ with $A_{δ} \subseteq δ$ such that every $A \subseteq ω_{1}$ has stationarily many $δ$ with $A \cap δ = A_{δ}$ ) and Jensen's square principles $□_{κ}$ hold, both flowing from condensation and the fine structure of $L$ ^{[Jech Ch. 13]}. $♢$ implies CH and builds a Suslin tree, so $L$ refutes the Suslin hypothesis. Against this richness stands the thinness of $L$ : if a non-identity elementary embedding $j : L \to L$ exists — equivalently the real $0^{#}$ exists — then $V \neq = L$ , every uncountable cardinal of $V$ is inaccessible in $L$ , and $L$ vastly undercomputes the universe; a measurable cardinal yields $0^{#}$ , so measurable and larger cardinals are flatly incompatible with $V = L$ . Jensen's covering theorem makes the dichotomy precise: either $0^{#}$ does not exist and $L$ approximates $V$ closely (every uncountable set of ordinals is covered by a constructible set of the same cardinality, so $L$ computes successors of singulars correctly), or $0^{#}$ exists and $L$ is a thin shadow.

Synthesis. The foundational reason a single inner model decides AC, GCH, $♢$ , and $□$ at once is the absoluteness of the constructible recursion: because $Def$ and the level function are computed identically in every inner model, $L$ is the least such model, it knows it is constructible ( $L ⊨ V = L$ ), and its definable global well-ordering makes choice a theorem. This is exactly the seam along which Gödel's relative consistency results run — $Con (ZF)$ buys $Con (ZF + V = L)$ , and $V = L$ proves AC and GCH, so $Con (ZF) \Rightarrow Con (ZFC + GCH)$ . The central insight is that condensation converts every elementary submodel of a level into a level, which generalises Löwenheim-Skolem [42.01] into rigidity: a constructible subset of $ω_{α}$ cannot escape stage $ω_{α + 1}$ , pinning $2^{ℵ_{α}}$ to $ℵ_{α + 1}$ and collapsing the gimel function of 42.03.04 to the successor. These results are dual to the forcing side of the chapter: where $L$ generates constraint by admitting only definable subsets, forcing generates freedom by adding generic subsets, making CH fail 42.03.07 and the regular continuum function nearly arbitrary (Easton, 42.03.04). Putting these together, $L$ and forcing are the two poles of independence: Gödel's $L$ shows GCH is consistent, Cohen's forcing shows its negation is consistent 42.03.08, and the bridge is that both read off the same continuum datum from opposite directions. The boundary is set by $0^{#}$ and the covering theorem, beyond which $L$ ceases to see the large-cardinal universe.

Full proof set Master

Proposition 1 (transitivity and ordinals of the levels). Each $L_{α}$ is transitive, $L_{α} \cap On = α$ , and $α \leq β \Rightarrow L_{α} \subseteq L_{β}$ .

Proof. Transitivity by induction: $L_{0} = \emptyset$ is transitive; if $L_{α}$ is transitive then every $a \in L_{α + 1} = Def (L_{α})$ has $a \subseteq L_{α} \subseteq L_{α + 1}$ , and any $y \in a$ lies in $L_{α} = Def (\dots)$ ? — more carefully, $y \in a \subseteq L_{α}$ , and $L_{α} \subseteq L_{α + 1}$ because the identity formula $y \in L_{α}$ -membership realises each $z \in L_{α}$ as ${w \in L_{α} : w \in z} \in Def (L_{α})$ using $z$ as a parameter, so $L_{α} \subseteq Def (L_{α}) = L_{α + 1}$ ; hence $y \in L_{α + 1}$ and $L_{α + 1}$ is transitive. Limits: a union of transitive sets is transitive, and $L_{α} \subseteq L_{λ}$ for $α < λ$ by the union. The monotonicity $L_{α} \subseteq L_{α + 1}$ just shown plus transfinite composition gives $α \leq β \Rightarrow L_{α} \subseteq L_{β}$ . The ordinal computation $L_{α} \cap On = α$ is Exercise 3. $□$

Proposition 2 ( $Δ_{0}$ -absoluteness and the absoluteness of $Def$ ). Every $Δ_{0}$ formula is absolute between transitive classes, and the relation " $d = Def (X)$ " is $Δ_{1}$ , hence absolute between transitive models of enough of ZF.

Proof. For $Δ_{0}$ -absoluteness, induct on formula complexity: atomic $x \in y$ and $x = y$ are absolute because they refer only to membership, which is the same relation in $M$ and $N$ ; Boolean connectives preserve absoluteness; a bounded quantifier $\exists x \in y ψ$ relativised to a transitive $M$ ranges over ${x \in M : x \in y} = {x : x \in y}$ since $y \in M$ and $M$ is transitive force $y \subseteq M$ , so the bounded range is the same in $M$ as in $N$ , and the induction hypothesis handles $ψ$ . For $Def$ : the satisfaction relation $Sat (X, ┌ φ ┐, \overset{a}{ˉ})$ for a set structure $(X, \in)$ is defined by recursion on the (finite) formula $φ$ , a recursion expressible by a $Σ_{1}$ formula (existence of a finite satisfaction-table) with a provably equivalent $Π_{1}$ form (uniqueness of the table), hence $Δ_{1}$ ; then $d = Def (X)$ asserts $d = {a \subseteq X : \exists φ \exists \overset{p}{ˉ} \in X a = {y \in X : Sat (X, ┌ φ ┐, (y, \overset{p}{ˉ}))}}$ , a $Δ_{1}$ condition. A $Δ_{1}$ relation is absolute between transitive models that verify the small fragment of ZF needed to run the satisfaction recursion. $□$

Proposition 3 (Lévy reflection for $L$ ). For any formula $φ (\overset{x}{ˉ})$ there are club-many $α$ such that $φ$ is absolute between $L_{α}$ and $L$ ; consequently Separation and Replacement hold in $L$ .

Proof. The levels $L_{α}$ form a continuous increasing chain with union $L$ , each $L_{α}$ transitive. The reflection theorem (cross-ref 42.03.01) applied to this chain yields, for the finitely many subformulas of $φ$ , a closed unbounded class of $α$ with $φ^{L_{α}} \leftrightarrow φ^{L}$ for all parameters in $L_{α}$ . For Separation: given $a \in L$ and a formula $φ$ with constructible parameters, choose $α$ with $a \in L_{α}$ and $φ$ reflecting; then ${y \in a : φ^{L} (y)} = {y \in a : φ^{L_{α}} (y)} \in Def (L_{α}) = L_{α + 1} \subseteq L$ . For Replacement: if $φ$ defines (in $L$ ) a function on a constructible $a$ , reflect $φ$ and the relevant ranges to a single $L_{α}$ containing the image, and apply Separation/Union inside $L$ to collect the image, which is then a member of some $L_{β}$ . $□$

Proposition 4 (Condensation, full statement). If $α$ is a limit ordinal and $M ≺ (L_{α}, \in)$ , then the transitive collapse of $M$ equals $(L_{β}, \in)$ for the ordinal $β = otp (M \cap On)$ , and $β \leq α$ .

Proof. By Mostowski (cross-ref 42.03.01) the well-founded extensional $(M, \in)$ collapses uniquely to a transitive $(N, π)$ with $π : M ≅ N$ . Since $V = L$ holds in $(L_{α}, \in)$ — every element of $L_{α}$ appears at a level below $α$ , and the level function is absolute (Proposition 2) so $L_{α}$ verifies "every set is some $L_{ξ}$ " — elementarity transfers $V = L$ to $M$ and the isomorphism transfers it to $N$ : $N ⊨ V = L$ . The level function being $Δ_{1}$ (Proposition 2) is absolute between the transitive $N$ and $V$ , so the object $N$ believes to be $L_{ξ}$ is the real $L_{ξ}$ for each $ξ \in N \cap On$ . As $N$ is transitive with $N ⊨ V = L$ , $N = ⋃ {L_{ξ} : ξ \in N \cap On} = L_{β}$ where $β = N \cap On = otp (M \cap On)$ (the collapse sends ordinals to their order type). Finally $β \leq α$ : $π$ does not increase ordinals (a collapse of a subset of $L_{α}$ has order type $\leq α$ ). $□$

Proposition 5 ( $L ⊨ GCH$ , full proof). Assuming the constructible recursion and Proposition 4, $L ⊨ 2^{ℵ_{α}} = ℵ_{α + 1}$ for all $α$ .

Proof. Work inside $L$ , so $V = L$ . Let $A \subseteq ω_{α}$ (so $A \in L$ ). Pick $γ > ω_{α}$ a limit ordinal with $A \in L_{γ}$ . Form the Skolem hull $M = Hull^{L_{γ}} (ω_{α} \cup {A})$ using a definable Skolem function for $(L_{γ}, \in)$ (available since $L_{γ}$ has a definable well-order $<_{L} ↾ L_{γ}$ , so least-witness Skolem functions exist without external choice); then $M ≺ L_{γ}$ , $ω_{α} \cup {A} \subseteq M$ , and $∣ M ∣ = ∣ ω_{α} ∣ = ℵ_{α}$ because the hull of a set of size $ℵ_{α}$ under countably many finitary functions has size $ℵ_{α}$ . By Proposition 4, $M$ collapses to some $L_{β}$ with $∣ L_{β} ∣ = ∣ M ∣ = ℵ_{α}$ , so $β < ω_{α + 1}$ . The collapse $π$ fixes $ω_{α} \cup {A}$ pointwise: each $ξ < ω_{α}$ is an ordinal below the least ordinal moved, and $A \subseteq ω_{α}$ is fixed because all its elements are; hence $A = π (A) \in L_{β} \subseteq L_{ω_{α + 1}}$ . Therefore $P (ω_{α}) = P (ω_{α}) \cap L \subseteq L_{ω_{α + 1}}$ , and $2^{ℵ_{α}} \leq ∣ L_{ω_{α + 1}} ∣ = ∣ ω_{α + 1} ∣ = ℵ_{α + 1}$ . Cantor gives $2^{ℵ_{α}} > ℵ_{α}$ , hence $\geq ℵ_{α + 1}$ , so equality. $□$

Proposition 6 (definable global well-ordering; $L ⊨ AC$ ). The relation $<_{L}$ is a parameter-free definable well-ordering of $L$ , and $L ⊨ AC$ .

Proof. By Proposition 1 the levels are increasing and transitive, and by Exercise 5 the level-by-level construction of $<_{L}$ (least defining pair, with Gödel number then $<_{L}$ -ordered parameter tuple) yields a linear order on each $L_{α + 1} ∖ L_{α}$ extending the order below; the union over levels is a linear order of all $L$ . It is a well-order: a nonempty constructible $S$ has a least level $β$ meeting it, and within $L_{β}$ the assigned defining pairs are well-ordered (Gödel numbers in $ω$ , tuples by induction), giving a $<_{L}$ -least element of $S$ . The defining formula for $<_{L}$ quantifies over levels, $Def$ , and Gödel numbering — all $Δ_{1}$ and absolute — so $<_{L}$ is definable without parameters and its definition is absolute to $L$ . Inside $L$ , given any family $F$ of nonempty sets, the function $X \mapsto (<_{L} -least element of X)$ is a choice function definable from $<_{L}$ ; hence $L ⊨ AC$ , and in fact $L ⊨$ "there is a definable well-ordering of the universe." $□$

Connections Master

Cofinality, cardinal exponentiation, and the singular cardinals hypothesis 42.03.04 is the direct prerequisite whose machinery this unit consumes and answers. That unit proves König's bound $cf (2^{κ}) > κ$ , the gimel function $ℷ (κ) = κ^{cf (κ)}$ , and the case analysis of $κ^{λ}$ , leaving the value of every exponential open; $L$ closes it completely, since $L ⊨ GCH$ forces $2^{ℵ_{α}} = ℵ_{α + 1}$ , collapses gimel to the successor, and makes the singular cardinals hypothesis a theorem. The cofinality refinement of 42.03.04 is exactly what the GCH count in $L$ respects: the captured stage $ω_{α + 1}$ is a regular cardinal, and condensation never violates the König constraint.
Forcing and generic extensions 42.03.07 is the co-produced companion that supplies the opposite pole of independence. Where $L$ admits only definable subsets and thereby proves GCH, forcing adds generic subsets of $ω$ to a ground model to refute CH, and the Easton product of 42.03.04 realises an arbitrary regular continuum function. This unit states that $L ⊨ GCH$ and is the least inner model; 42.03.07 builds the generic extensions that move outward from such a ground model, and the two constructions are dual — inner model versus outer extension, definability versus genericity.
The independence of the continuum hypothesis 42.03.08 is the co-produced unit where the two halves meet: Gödel's $L$ (this unit) gives $Con (ZFC) \Rightarrow Con (ZFC + CH)$ via GCH, and Cohen's forcing (42.03.07) gives $Con (ZFC) \Rightarrow Con (ZFC + \neg CH)$ , so CH is independent of ZFC. The Axiom of Choice 42.03.05 is settled the same way: its consistency is Theorem 3 here (via $L ⊨ AC$ ), its negation's consistency is the symmetric-extension result cross-referenced from 42.03.05, and the present unit is the definability half of both independence results.

Historical & philosophical context Master

Kurt Gödel announced the relative consistency of the axiom of choice and the generalised continuum hypothesis in two notes to the Proceedings of the National Academy of Sciences in 1938 and 1939, with the full construction in his 1940 Princeton monograph The Consistency of the Continuum Hypothesis ^{[Gödel 1940]}. He defined the constructible sets by transfinite recursion through the eight fundamental operations now called the Gödel operations, which generate at each successor stage exactly the first-order definable subsets of the previous level, and proved that the resulting class $L$ models ZF together with $V = L$ , AC, and GCH. The result was the first half of the independence of CH: it showed CH cannot be refuted from ZFC, leaving open whether it could be proved — a question settled negatively by Paul Cohen's 1963 forcing construction, the two together establishing full independence.

The structural depth of $L$ emerged later. Ronald Jensen's fine-structure theory of the 1970s extracted from condensation the combinatorial principles $♢$ and $□$ , used to build Suslin trees and to control reflection at singular cardinals; Jensen's covering theorem then drew the sharp dichotomy governed by $0^{#}$ , the remarkable set of indiscernibles for $L$ isolated by Jack Silver and Robert Solovay around 1966-1970 whose existence (a consequence of a measurable cardinal, by Dana Scott's 1961 theorem that a measurable contradicts $V = L$ ) forces $V \neq = L$ and makes $L$ a thin shadow of the universe. The constructible universe thus occupies a precise place: the minimal model in which the classical hypotheses are decided affirmatively, and the canonical object against which the large-cardinal hierarchy measures its distance from definability.

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}
}

@book{godel1940consistency,
  author    = {G{\"o}del, Kurt},
  title     = {The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory},
  series    = {Annals of Mathematics Studies},
  volume    = {3},
  publisher = {Princeton University Press},
  year      = {1940}
}

@article{godel1938pnas,
  author  = {G{\"o}del, Kurt},
  title   = {The consistency of the axiom of choice and of the generalized continuum-hypothesis},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {24},
  year    = {1938},
  pages   = {556--557}
}

@book{devlin1984constructibility,
  author    = {Devlin, Keith J.},
  title     = {Constructibility},
  series    = {Perspectives in Mathematical Logic},
  publisher = {Springer},
  year      = {1984}
}

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

@incollection{jensen1972fine,
  author    = {Jensen, Ronald B.},
  title     = {The fine structure of the constructible hierarchy},
  journal   = {Annals of Mathematical Logic},
  volume    = {4},
  year      = {1972},
  pages     = {229--308}
}

@article{silver1971zerosharp,
  author  = {Silver, Jack H.},
  title   = {Some applications of model theory in set theory},
  journal = {Annals of Mathematical Logic},
  volume  = {3},
  year    = {1971},
  pages   = {45--110}
}

Prerequisites

42.03.04

Tier anchors

beginner: Halmos 1960 *Naive Set Theory* (Van Nostrand) §11-13 (the cumulative hierarchy and definable subsets read by hand); Hrbacek-Jech 1999 *Introduction to Set Theory* 3e (Marcel Dekker) Ch. 14 (the axiom of constructibility and the consistency results for an informal first pass)
intermediate: Kunen 2011 *Set Theory* (College Publications) Ch. II (the constructible universe: the function Def, the hierarchy L_α, the relativisation of formulas, absoluteness for transitive models, L ⊨ ZF, the condensation lemma, V=L, and the proofs that L ⊨ AC and L ⊨ GCH); Jech 2003 *Set Theory* 3e (Springer) Ch. 13 (inner models and constructibility)
master: Kunen 2011 *Set Theory* (College Publications) Ch. II §1-6 and the exercises on relative constructibility and the fine structure pointers, with Ch. IV for the absoluteness preliminaries; Jech 2003 *Set Theory* 3e (Springer) Ch. 13 (constructibility, the condensation lemma, GCH and ◊ in L), Ch. 18 (the covering theorem and 0#), and Ch. 27 (◊ and □); Devlin 1984 *Constructibility* (Springer Perspectives in Logic) for the fine-structure development

References

Kunen, K. — Set Theory · College Publications, Studies in Logic 34 (2011, revised edition). Chapter II develops Gödel's constructible universe. For a set X the definable power set Def(X) is the collection of subsets of X defined by a first-order formula with parameters from X interpreted in the structure (X, ∈); the constructible hierarchy is L_0 = ∅, L_{α+1} = Def(L_α), L_λ = ⋃_{α<λ} L_α at limits, and L = ⋃_{α ∈ On} L_α. Kunen sets up the relativisation φ^M of a formula to a class M and the Lévy hierarchy Σ_0 = Π_0 = Δ_0 (bounded quantifiers), Σ_1, Π_1, …, proving the Δ_0-absoluteness of bounded formulas between transitive models and the upward/downward absoluteness of Σ_1/Π_1 formulas; ordinals, ω, transitivity, the pairing/union/power-set relations, rank, and the satisfaction relation for set structures are shown absolute or Δ_1, while cardinality, the power set, and 'x is constructible' are not absolute in general. An inner model is a transitive class containing all ordinals and modelling ZF; L is shown to be the least inner model — L ⊆ M for every inner model M — by verifying each ZF axiom holds in L (using that Def is absolute enough and that the L_α are uniformly definable) and that the defining recursion is absolute. The condensation lemma states that every elementary substructure of (L_α, ∈) has transitive collapse (L_β, ∈) for some β ≤ α; combined with the Skolem-hull/Mostowski-collapse method it yields L ⊨ V=L, L ⊨ AC via the definable global well-ordering <_L recursively built level by level, and L ⊨ GCH: a subset A ⊆ L_{ω_α} that is constructible appears in some L_γ, and condensation collapses a hull of size ℵ_α containing A to some L_β with β < ω_{α+1}, so A ∈ L_{ω_{α+1}} and 2^{ℵ_α} ≤ |L_{ω_{α+1}}| = ℵ_{α+1}. Hence Con(ZF) ⟹ Con(ZFC + GCH), Gödel's 1938 relative consistency theorem. The combinatorial principles ◊ and □ are shown to hold in L (Jensen), and the chapter notes the limits of L: the existence of 0# implies V ≠ L and that measurable cardinals contradict V=L, so L is a 'thin' inner model insensitive to large cardinals at the level of measurability and above.
Jech, T. — Set Theory · Springer Monographs in Mathematics, 3rd millennium edition (2003). Chapter 13 develops constructibility and inner models: the definable power set Def(M), the hierarchy L_α and the class L, the relativised satisfaction and the Gödel operations giving an alternative finite-rudimentary generation of Def (so the recursion is Σ_1 over the structure and L_α is uniformly Δ_1 definable from α), the proof that L is an inner model of ZF and is the smallest such (⊆ every inner model), the axiom of constructibility V=L and its absoluteness L ⊨ V=L (so (V=L)^L holds), the global well-ordering <_L definable without parameters making L ⊨ AC, and the condensation lemma (elementary submodels of L_α collapse to L_β) yielding L ⊨ GCH and the regularity that constructible subsets of ω_α appear by stage ω_{α+1}. Chapter 13/15 derive the combinatorial principles ◊_κ (diamond) and □_κ (square) in L from condensation and fine structure (Jensen), used to build Suslin trees and to refute compactness at successors of singulars; Chapter 18 develops the limits of L — the covering theorem (Jensen): if 0# does not exist then L computes successors of singular cardinals correctly and is 'close' to V, whereas 0# (a non-identity elementary embedding L → L, equivalently a remarkable set of indiscernibles for L) implies V ≠ L decisively, and a measurable cardinal yields 0#, so measurable and larger cardinals are incompatible with V=L. The relative consistency Con(ZF) ⟹ Con(ZFC + GCH) (Gödel 1938) and the contrast with Cohen forcing (which makes CH fail) are the headline theorems.
Gödel, K. — The Consistency of the Continuum Hypothesis · Annals of Mathematics Studies 3, Princeton University Press (1940), based on the 1938-1939 Princeton lectures and the two 1938/1939 *Proceedings of the National Academy of Sciences* announcements (vol. 24, 556-557 and vol. 25, 220-224). Gödel defines the constructible sets by transfinite recursion using the eight fundamental (Gödel) operations to generate, at each successor stage, exactly the subsets of the previous level first-order definable over it, sets L = ⋃_α L_α, and proves that L is a model of ZF in which the axiom of choice and the generalised continuum hypothesis hold. The proof of AC is the construction of a definable well-ordering of L following the order of generation; the proof of GCH rests on the condensation argument that any constructible set of constructible subsets of an aleph ℵ_α is generated by the stage ℵ_{α+1}, bounding 2^{ℵ_α} ≤ ℵ_{α+1}. The monograph establishes Con(ZF) ⟹ Con(ZF + AC + GCH), settling the relative consistency (not provability) of AC and GCH and forming, with Cohen's later independence proof, the two halves of the independence of CH from ZFC.

Estimated time

beginner: 20m
intermediate: 50m
master: 90m