Title:

Some topics in model theory.

This thesis is concerned with various topics from first
order model theory. In chapter 1, we prove that every consistent
sentence from a language with no function symbols and two
variables has a finite model. Results concerning the spectrum
of such a sentence, as tjell as a decidability result are shown
to follow.
The concept of a strongly minimal formula is introduced in
chapter 2 where, by considering a certain "algebraic" property
of theories, we strengthen for strongly minimal theories some
known results applying more generally.
Chapter 3 is concerned with carstable theories having rank
of transcendance 2. These theories are first characterised in
terms of strongly minimal formulae, and this characterisation
then used to prove that every universal model of such a theory
is saturated. Some general considerations arising from this
result are also given, as well as a proof for the theories considered4of
a conjecture of Lachlan. Along the way, we give our
own proof of Baldwin's result that every
A,
categorical theory
of rank 2 is almost strongly minimal.
In chapter 4 we look at the question of how nearly model
complete are i1categorical theories. This question has been
given two precise formulations by Macintyre, both of which we
answer. We introduce and investigate the notion of a theory
being "almost model complete", and by showing that certain
finitely axiomatisable theories have this property, answer a
question of Dickmann on the finite axiomatisability of certain
}{ýcategorical theories.
The second part of this thesis is concerned with modal logic.
In chapter 5 we show that for many logics, compactness and the
LowenheimSkolem theorems follow immediately from weak complete
ness. Other completeness results are proved, as well as an
omitting types theorem. Finally, in chapter 6 we characterise
for various logics those sentences preserved under various notions
of extension.
