Summary: The Index Set of Uncountably Categorical Theories
Uri Andrews, Tamvana Makuluni
July 7, 2011
We classify the complexity of the index set of uncountably categorical theories. We
show that this index set surprisingly falls at the intermediate stage of being complete
for intersections of 2 sets with 2 sets.
One goal of mathematical logic is to determine the complexity of mathematical notions.
The methods most often used to measure complexity of a notion are Kleene's arithmetical
and analytic hierarchies. A set is n if it can be described in arithmetic by a formula with
n quantifiers beginning with an existential quantifier. A set is n if it is the complement
of a n set. A set is arithmetical (classifiable in Kleene's arithmetical hierarchy) if it is
n for some n. The most natural characterizations of uncountable categoricity are non-
arithmetical, leading to the question of whether uncountable categoricity is an arithmetical
notion, and if so of which complexity. To completely characterize the complexity of a
set, we show that it is complete at some level of the hierarchy. Lempp and Slaman
 characterize the complexity of 0-categoricity and Ehrenfeuchtness, two other natural
model theoretic notions, showing that one is arithmetical and 3-complete, while the other
is non-arithmetical. Most natural mathematical notions are n-complete or n-complete.