Building L Examples

Set Theory · Axiom Academy

EXAMPLE Building the Constructible Universe Walk through the construction of L₀, L₁, L₂, and L₃ step by step, seeing how definability restricts what sets appear at each level. Excellent work! You've explored how the constructible universe L is built. Here's what we learned: Starting point: L₀ = ∅, we begin with nothing but the empty set Definability restriction: At each stage, Def(Lₐ) contains only subsets of Lₐ that can be defined by first-order formulas Gradual growth: L₁ = ∅ , L₂ = ∅, ∅ , L₃ = ∅, ∅ , ∅ , ∅, ∅ — the universe grows systematically The full picture: L = ⋃ₐ Lₐ where α ranges over all ordinals, creating a minimal model of ZF The constructible universe L is the "smallest" possible model of set theory, containing only sets that can be explicitly defined!

This is the written version of the interactive lesson above. See the full Set Theory course.