Knowlpedia

Definition

Hierarchy of parameter choices

An ordered choice of small or large parameters that makes the permitted dependencies explicit.

A hierarchy of parameter choices specifies which constants are fixed before later ones are selected. A statement such as 0<α≪β≪10<\alpha\ll\beta\ll1 is shorthand, not a numerical relation: first choose β\beta sufficiently small for the fixed data, then choose α\alpha sufficiently small depending on those data and on β\beta.

Quantifier order

One precise form is

∃β∗>0  ∀β∈(0,β∗)  ∃α∗(β)>0  ∀α∈(0,α∗(β)): P(α,β).\exists\beta_*>0\;\forall\beta\in(0,\beta_*)\; \exists\alpha_*(\beta)>0\;\forall\alpha\in(0,\alpha_*(\beta)):\ P(\alpha,\beta).

The forbid choosing β\beta in response to α\alpha. Large parameters can be handled by making their reciprocals small. A proof must exhibit or justify compatible thresholds; the symbol ≪\ll alone supplies none.