84163 - Model Theory (1) (LM)

Course Unit Page

Academic Year 2018/2019

Learning outcomes

The course aims to provide a basic knowledge in Model Theory, a branch of Mathematical Logic treating the interactions connecting logical language and mathematical models, with particular emphasis on the formulation of formal axiomatic systems dealing with mathematical structures. At the end of the course the student will be acquainted with some of the main research fields in the subject and with their corresponding foundational results.

Course contents

Model Theory is a branch of Logic concerning the relationships that join logical language and mathematical structures. More precisely, it uses the logical language as an applied tool for mathematical investigations. Logicians, in fact, do not use rulers or compasses for “doing mathematics”, but rather the language itself, by which they describe the mathematical models they want to investigate.

What kind of relationships connect then syntactic theories with underlying mathematical structures? How do the properties satisfied by the earlier reflect in those fulfilled by the others? For instance, first order theories whose validity is preserved from models to their substructures can be axiomatized by using universal formulas only. Another example: mathematical theories satisfying certain algebraic properties admit the “quantifier elimination”, that is, every formula in their language is equivalent within the theory to some quantifier-free formula. If moreover a mechanical method exists to determine the truth of closed quantifier-free formulas, then the theory in question is decidable. Model Theory is hence a branch of Logic whose results can increase considerably our knowledge of mathematical structures (and remarkable applications have in fact occurred in algebraic geometry for the solution of important conjectures, such as the Mordell-Lang conjecture).

The purpose of this course is that of introducing the basic notions of Model Theory according to the following directions: Tarskian semantics for predicate logic; fundamental algebraic structures; Compactness Theorem and its consequences; universally axiomatized theories: monoids, groups, rings; categorical theories: linear dense orders without extrema and torsion-free Abelian groups; elimination of quantifiers: linear dense orders without extrema and algebraic closed fields. Decidable theories: Vaught test, linear dense orders without extrema and algebraic closed fields.

As for the necessary theoretical prerequisites, ordinary school algebraic basic notions suffice together with the usual logical background provided by any first level Logic course.

Readings/Bibliography

The fundamental didactic material will be provided by the teacher. As side support material the following books are recommended:

A. Marcja, C. Toffalori: Introduzione alla teoria dei modelli, Pitagora, 1998

D. Marker: Model theory: an introduction, Springer, 2002

Teaching methods

Lessons in classrooms with blackboard.

Assessment methods

The final exam will consist in an oral test, in which students are supposed to prove their correct comprehension of the notions dealt with during the lessons, by their oral explanation but also by written exposition of the fundamental definitions and results of the subject. Moreover, it is required the capacity of reconstructing the proofs concerning:

*) a selected topic among 1) Compactness Theorem and its corollaries 2) universally axiomatizable theories 3) categoricity

(Further possibile choices can be agreed with the teacher)

*) quantifier elimination: general elimination procedure and proofs for a specific case (Theory of Dense Strict Linear Order without Extremes or Closed Algebraic Fields)

*) decidability of first-order theories.

Teaching tools

Lessons will be held by using a blackboard.

Typewritten didactic material will be provided by the teacher.

Office hours

See the website of Guido Gherardi