 
Summary: QUANTIFIER RANK SPECTRA OF SCATTERED
SENTENCES OF L1,
BY NATHANAEL LEEDOM ACKERMAN
Abstract. In this paper we show that, assuming the existence of the
ories with a nice metalanguage, there is a set Z, unbounded in 1, such
that for every Z there is a scattered sentence S L1, where the
quantifier rank of S is less than or equal to but the supremum of the
quantifier ranks of countable models of S is .
1. Introduction
In this paper we study the possible quantifier rank spectra of sentences
of L1, under the assumption that a nice metalanguage exists. We show
that under these assumptions there is a set Z of ordinals unbounded in 1
such that for each Z there exists a scattered sentence S L1, with
quantifier rank less than or equal to but with quantifier rank spectrum
unbounded in .
We begin in Section 2 by introducing the theory of infinitely branch
ing trees. In Section 3 we then discuss some more elaborate tree structures.
These will include structures which allows us to compare the heights of trees
as well as what we call full trees.
In Section 4 we introduce the concept of base predicates and of archetypes.
