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Reflection principles assert that any property true in the universe V is true in some initial segment V_α. The axiom of replacement and infinity are both reflection-type axioms. Stronger reflection principles (not provable in ZFC) postulate that V is 'indescribable,' implying the existence of large cardinals. Reflection bridges V's vastness with the approachability of its fragments.
Prove basic reflection: for any formula φ, there exists α such that φ is true in V_α iff it is true in V (by induction on formulas). Explain how measurability can be phrased as a reflection principle. Introduce supercompact and strongly inaccessible cardinals as reflection strengths.
To understand reflection, start with what you already know about the cumulative hierarchy: V is built by iterating the power set operation through all ordinal stages, so V = ∪{V_α : α ∈ Ord}. Each V_α is a set — a bounded, surveyable fragment of the entire universe. The key question reflection asks is: how much of what is true in V is already "visible" inside some V_α? The reflection principle answers that any statement true in V was already true in some initial segment.
More precisely, for any first-order formula φ(x₁, …, xₙ) and any set M in V, there exists an ordinal α large enough that V_α contains M and φ holds in V exactly when it holds in V_α (with the same witnesses). You already know from elementary submodels that an elementary submodel M ≺ V satisfies the same first-order sentences as V. Reflection is the ordinal-indexed version of this: the cumulative hierarchy produces, for each formula, cofinally many stages that are elementary submodels of V for that formula. ZFC itself proves this (using replacement and the fact that the hierarchy is indexed by all ordinals), so basic reflection is a theorem, not an additional axiom.
The philosophically rich move comes when you push reflection beyond what ZFC can prove. The idea is that the universe V is so vast it cannot be "pinned down" by any single property — whatever you can say about V using a large cardinal axiom must already hold at some set-sized cardinal below. This is indescribability: a cardinal κ is strongly inaccessible if you can't "describe it away" with a property; a measurable or supercompact cardinal κ is one that reflects even more complex properties. Each step up the large cardinal hierarchy corresponds to a stronger reflection principle — a claim that V mirrors itself downward in a more complete way.
A subtle trap worth avoiding: reflection is not the same as the Löwenheim-Skolem theorem. Löwenheim-Skolem gives you a countable elementary substructure of any structure — a purely logical result about first-order theories. Reflection is set-theoretic: it uses ordinal indexing and the specific structure of the cumulative hierarchy in V. The submodels reflection produces are transitive (they are honest V_α's) and cofinally many, not arbitrary countable structures. Transitivity is what makes them useful for set-theoretic arguments — elements of a transitive model are themselves sets in V with all their ∈-relations intact.
The practical payoff is that reflection lets you "localize" arguments about V. To show a property holds of some large cardinal, it suffices to show V reflects it. To prove relative consistency of large cardinal axioms from each other, one constructs inner models that inherit the reflection properties of V. The phrase "builds toward inner models and consistency strength" in this topic's metadata points exactly here: reflection is the engine that drives the large cardinal hierarchy, because each axiom is essentially the claim that some form of reflection holds at a particular cardinal.