Summary: On the Relationship Between ATR0 and ID<
April 2, 1996
We show that the theory ATR0 is equivalent to a second-order general-
ization of the theory cID<. As a result, ATR0 is conservative over cID<
for arithmetic sentences, though proofs in ATR0 can be much shorter than
their cID< counterparts.
Let 0 denote the least impredicative ordinal, as defined in  or . Work of
Feferman and Sch¨utte in the sixties demonstrated that this is the proof-theoretic
ordinal corresponding to theories embodying "predicative mathematics." In
more recent years a number of classical theories without direct predicative jus-
tification have been discovered, whose proof-theoretic strength is also 0. The
aim of this paper is to clarify the relationship between two such theories, namely
ID< and Friedman's ATR0.
The IDn are theories in the language of Peano Arithmetic augmented by
new constants representing fixed points of arithmetic formulas involving positive
occurences of a unary predicate. Each theory IDn allows n iterations of this