Let T be a consistent r.e. extension of Peano arithmetic;
Σ
n
0
\Sigma _n^0
,
Π
n
0
\Pi _n^0
the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting “for free"); and
Γ
\Gamma
,
Γ
′
\Gamma ’
variables through the set of all classes
Σ
n
0
\Sigma _n^0
,
Π
n
0
\Pi _n^0
. The principal concern of this paper is the question: When can we find an independent sentence
ϕ
∈
Γ
\phi \, \in \,\Gamma
which is
Γ
′
\Gamma ’
-conservative in the following sense: Any sentence
χ
\chi
in
Γ
′
\Gamma ’
which is provable from
T
+
ϕ
T + \phi
is already provable from T? (Additional embellishments: Ensure that
ϕ
\phi
is not provably equivalent to a sentence in any class “simpler” than
Γ
\Gamma
; that
ϕ
\phi
is not conservative for classes “more complicated” than
Γ
′
\Gamma ’
.) The answer, roughly, is that one can find such a
ϕ
\phi
, embellishments and all, unless
Γ
\Gamma
and
Γ
′
\Gamma ’
are so related that such a
ϕ
\phi
obviously cannot exist. This theorem has applications to the theory of interpretations, since “
ϕ
\phi
is
Γ
\Gamma
-conservative” is closely related to the property “
T
+
ϕ
T + \phi
is interpretable in T"-or to variants of it, depending on
Γ
\Gamma
. Finally, we provide simple model theoretic characterizations of
Γ
\Gamma
-conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).