The Constructible Universe L

Set Theory · Axiom Academy

LESSON The Constructible Universe L Gödel's inner model built from definable sets only 1. Building from Definable Sets Only The constructible universe L is built similarly to V, but we only include sets that are definable from parameters in earlier stages. We write Def(M) for the collection of all sets definable over M. This is the key operation in building L. We build L by transfinite recursion, analogous to V but using Def instead of P: Since Def(M) ⊆ P(M) for any M, we have Lα ⊆ Vα for all ordinals α, and therefore L ⊆ V. This makes L an inner model of ZFC. The statement "V = L" (every set is constructible) can be added as an axiom to ZFC. While this axiom is not generally accepted as true, it provides a useful framework for studying set theory. If we assume V = L, then we're restricting our universe to only those sets that can be explicitly defined. This eliminates many "exotic" sets that might exist in other models. The constructible universe has several remarkable properties that make it special: L ⊨ ZFC - L is a model of all ZFC axioms L ⊨ AC - The Axiom of Choice holds in L L ⊨ GCH - The Generalized Continuum Hypothesis holds in L L is minimal - L ⊆ M for any inner model M of ZFC Absoluteness - Many properties are absolute between V and L These properties allow Gödel to prove: If ZFC is consistent, then so is ZFC + AC + GCH.

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