Axiom of constructibility

The axiom asserting that every set belongs to Gödel’s constructible universe L, written V = L. It strengthens ZFC by restricting the universe of sets to those obtainable through a definable hierarchy.

Generalized continuum hypothesis: The assertion that every infinite cardinal κ satisfies 2^κ = κ⁺. V = L implies this cardinal-arithmetic principle, though ZFC alone cannot decide it.
Constructible universe: The class L formed by iterating definable subsets through the ordinals. V = L says that this internally generated class contains every set.
Kurt Gödel: An Austrian-American logician whose incompleteness theorems transformed mathematical logic. Gödel developed L and proved the relative-consistency results underlying the axiom.
Forcing (mathematics): A method for constructing extensions of models of set theory to establish independence results. Forcing produces models where propositions such as CH fail, unlike the consequences of V = L.
Axiom of choice: The principle that every family of nonempty sets has a choice function. The constructible universe satisfies choice, so V = L implies the axiom of choice.
Definable subset: A subset specified by a formula with parameters from a given structure. At successor stages, L adds subsets definable over the preceding stage.
Continuum hypothesis: The assertion that no cardinal lies strictly between the cardinality of the integers and the real numbers. Gödel’s constructible universe proof showed that CH holds in a model of ZF.
Cohen's independence proof: Paul Cohen’s proof that the continuum hypothesis and axiom of choice are independent of ZF, assuming consistency. Cohen’s method complements Gödel’s inner-model argument by showing CH can fail in an extension.
Suslin hypothesis: The assertion that every dense linear order without endpoints satisfying the countable chain condition is separable. V = L implies the Suslin hypothesis, ruling out Suslin lines.
Cumulative hierarchy: The hierarchy Vα of sets built by iterated power-set formation along the ordinals. V = L identifies the full cumulative universe with a more selective hierarchy.

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