 
Summary: An ordinal analysis of admissible set theory
using recursion on ordinal notations
Jeremy Avigad
May 23, 2001
Abstract
The notion of a function from N to N defined by recursion on ordinal
notations is fundamental in proof theory. Here this notion is generalized to
functions on the universe of sets, using notations for wellorderings longer
than the class of ordinals. The generalization is used to bound the rate
of growth of any function on the universe of sets that is 1definable in
KripkePlatek admissible set theory with an axiom of infinity. Formalizing
the argument provides an ordinal analysis.
1 Introduction
In informal prooftheoretic parlance, the definition of a set of objects is said to
be impredicative if it makes reference to a collection of sets that includes the
set being defined. A classic example arises if one takes the real numbers to be
lower Dedekind cuts of rationals, and then defines the least upper bound of a
bounded set of reals to be the intersection of all the upper bounds. A theory is
said to be (prima facie) impredicative if its intended interpretation depends on
such a definition.
