Some topics in model theory

• Main Author: Mortimer, M.
• Published: Royal Holloway, University of London 1974
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.466361

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 car-stable 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 i1-categorical 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 }{ý-categorical theories. The second part of this thesis is concerned with modal logic. In chapter 5 we show that for many logics, compactness and the Lowenheim-Skolem 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.