Course Title: (A4) Elementary Model Theory

[ For third year philosophy undergraduates and above, Credits: 3, Fall, 2003 ]

Model theory is in essence a branch of mathematical logic, which deals with, roughly speaking, the relationship between the formal language in use (in particular, suitable for a formal logical system, or a mathematical theory) and models (structures) which satisfy sets of sentences of the language in use. Emphasizing on the mathematical aspect, model theory is the study of the constructions of structures which realise mathematic theories, e.g. analysis, algebra, ... etc. At the core of this introductory course, we shall mainly concerned with the construction of structures for formal language in use ( or just sets of sentences of a fixed language, especially a first-order one), and the relationships among structures.

The course is intended for third year philosophy and mathematics undergraduates and above who are specially interested in formal logic and who may intend to do further study in the philosophy of language and logic, and of course the philosophy of mathematics and science. The burden of of this introductory course is to provide a brief understanding of constructing models of first-order logic. An easy starting point will be the construction of a Henkin-style proof of the completeness of the predicate calculus. Two introductory courses (namely, Elementary Logic and Elementary Set Theory) are usually assumed, though not so essential. However, I shall assume students` familiarity with some basic concepts, such as
The establishment of a first-order language suitable for the predicate calculus
The construction of structures appropriate for the established first-order language
The concepts of semantic sequents and tautologies/valid formulae
The presentation of a formal system for predicate logic - axioms, rules of inference, derivations and theorems
Some fundamental meta-theoretical results, such as general entailment relations, soundness and completeness
Basic properties of sets
Axioms of Zermelo-Fraenkel set theory
Cardinals, axiom of choice, Zorn`s lemma and well-ordering principle
Countable and uncountable sets
Isomorphism of ordered sets
Ordinals and ordinal types.
Since this introductory course is mainly for philosophy students, I shall ignore most of topics and examples involving mathematical study such as groups, algebra, ... etc., but concentrate on the general properties of first-order models and the methods of how to construct a desired model.

Suggested textbooks

J. Bridge, Beginning Model Theory (Oxford, Clarendon Press, 1977)
C. C. Chang and H.J. Keisler, Model Theory (3rd ed., North-Holland, 1990, chs 1-3 and 4.1).
W. Hodges, Shorter Model Theory, Cambridge University Press, 1997.
H.J. Keisler, `Fundamentals of Model Theory `, Chapter A.2 of Handbook of Mathematical Logic, J. Barwise (ed.), North Holland, 1977.