  The central result of this paper is the small-is-very-small principle for
restricted sequential theories. The principle says roughly that whenever the
given theory shows that a property has a small witness, i.e. a witness in every
definable cut, then it shows that the property has a very small witness: i.e. a
witness below a given standard number.
  We draw various consequences from the central result. For example (in rough
formulations): (i) Every restricted, recursively enumerable sequential theory
has a finitely axiomatized extension that is conservative w.r.t. formulas of
complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension
that is elementary for formulas of complexity $\leq n$, in which the
intersection of all definable cuts is the natural numbers. (iii) We have
reflection for $\Sigma^0_2$-sentences with sufficiently small witness in any
consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable
and sequential. Suppose further that every recursively enumerable and
sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$
is mutually globally interpretable with a finitely axiomatized sequential
theory.
  The paper contains some careful groundwork developing partial satisfaction
predicates in sequential theories for the complexity measure depth of
quantifier alternations.
