 
Summary: Quantifier Rank Spectrum of
L, (PhD Thesis Defense)
by Nathanael Leedom Ackerman
April 19, 2006
Definition 1. If L is a relational language then
L1,(L) is the smallest collection of formulas
such that if (x) L1,(L) then
· L L1,(L)
· ¬(x) L1,(L)
· (y)(x) L1,(L)
· (y)(x) L1,(L)
and if {i(x) : i } L1,(L) where iFree
Variables(i) is finite then
· i i(x) L1,(L)
· i i(x) L1,(L)
1
Definition 2. If L is a relational language and
(x) L1,(L) we define the quantifier rank
of (x) (qr((x))) by induction:
· qr(R(x)) = 0 if R is a relation in L.
