 
Summary: On the Relationship Between ATR0 and ID<
Jeremy Avigad
April 2, 1996
Abstract
We show that the theory ATR0 is equivalent to a secondorder 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.
1 Introduction
Let 0 denote the least impredicative ordinal, as defined in [9] or [12]. Work of
Feferman and Sch¨utte in the sixties demonstrated that this is the prooftheoretic
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 prooftheoretic 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
