Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use
$[\lambda |X]_T^{\rm{\Lambda }}\varphi$
as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.
For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then
$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$
In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).
Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion,
${\rm{RCA}}_0^{\rm{*}}$
be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,
$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$
We may even replace
${\rm{RCA}}_0^{\rm{*}}$
by the weaker ECA0, the second-order analogue of Elementary Arithmetic.
Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.