Lévy reflection theorem (ZFC)
Statement
In ZFC, for every finite list of formulas in the language {∈, =}, there are arbitrarily large ordinals α such that each formula reflects between the structure (V_α, ∈) and the universe (V, ∈) when parameters lie in V_α—that is, truth in the rank V_α matches truth in V for those formulas. Intuitively, any finite fragment of first-order set theory already holds in an initial segment of the cumulative hierarchy. This is a theorem proved from the ZFC axioms (the axiom schema of Replacement is essential); it is not assumed as a primitive axiom of ZFC. Stronger “reflection principles” beyond this finite form are sometimes added as extra axioms in extended theories.
Formula
∀ finite Φ ⊆ L_{∈} ∀α ∃β>α (V_β, ∈) ⊧ φ[ā] ⇔ (V, ∈) ⊧ φ[ā] for φ ∈ Φ, ā ∈ V_β
Proof sketch
Standard proof by transfinite induction / model-theoretic argument using the cumulative hierarchy V_α; Replacement is used to define Skolem hulls or analogous closures. See set-theory textbooks (e.g. Kunen, Jech). Closely related formulations are due to Montague (1961) and Lévy (1965).
Source
Lévy, A. A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, No. 57, 1965. Montague, Fraenkel’s addition to the axioms of set theory, in Essays on the foundation of mathematics, 1961.
Domains
mathematics