Constructible Universe

Origin. Gödel (1938). The inner model L, built by iterating first-order definability. Established the consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with ZF.

Models. L is the cumulative hierarchy admitting only definable subsets at each stage. It is the minimal inner model of ZF and satisfies its own defining axiom V = L.

Formalism.

Definable power set: Def(X) = subsets of X definable with parameters over (X, ∈).

Hierarchy: L₀ = ∅. L_{α+1} = Def(L_α). L_λ = ⋃{α<λ} L_α (λ limit). L = ⋃{α ∈ Ord} L_α.

Gödel's theorem: L ⊨ ZFC + GCH + (V = L). Choice and GCH hold in L.

Condensation lemma: Elementary submodels of L_α collapse to some L_β. Drives GCH and the diamond principle ◊.

Global structure: L has a definable Σ₁ well-ordering. Fine structure (Jensen): ◊, ☐ principles hold.

Large cardinals: V = L excludes measurable and larger cardinals. 0# does not exist under V = L.

Symbols.

Symbol	Unicode	Name	Meaning
L	—	Constructible universe	Inner model
L_α	—	Level α	Stage of hierarchy
Def	—	Definable power set	Definable subsets
V=L	—	Axiom	Everything constructible
◊	U+25C7	Diamond	Combinatorial principle

Metatheory. L is an absolute, definable inner model of every model of ZF. Con(ZF) ⟹ Con(ZFC + GCH). Paired with forcing (which gives Con(ZFC + ¬CH)), it establishes the independence of CH. Fine-structure theory yields powerful combinatorics.

Applies to. Relative-consistency proofs. Inner model theory. Descriptive set theory (under V = L). Cardinal arithmetic.

Limitations. V = L decides much but contradicts large cardinals, so it is not adopted as a foundational truth. A consistency tool, not a claim about the universe. Restrictive combinatorics.