# 2019/20 Taught Postgraduate Module Catalogue

## MATH5120M Advanced Models and Sets

### 20 creditsClass Size: 45

Module manager: Dr Vincenzo Mantova
Email: V.L.Mantova@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisites

 MATH2040 Mathematical Logic 1

### This module is mutually exclusive with

 MATH3120 Models and Sets

This module is approved as an Elective

### Module summary

Set theory is generally accepted as a foundation for mathematics, in an informal sense. It is also a formal axiomatic system, as developed by Zermelo and Fraenkel, among others, building on work of Cantor. Model theory is the study of formal axiomatic systems in full generality, and also depends on set theory for many of its basic definitions and results. Model theory and set theory constitute two of the basic strands of mathematical logic. They present rather special ways of viewing different parts of mathematics from a common perspective. In this module we explain the basic notions of these interrelated subjects.We will discuss in addition some specialized and advanced topics, which may vary.

### Objectives

On completion of this module, students should be able to:
a) test various abstraction terms for sethood;
b) use set theory to set up a foundation for mathematics, including constructions of some basic number systems;
c) handle elementary arguments involving ordinals and cardinals;
d) understand the axiom of choice;
e) describe the relationships between first order languages and structures, and understand the proof of the compactness theorem of first order logic;
f) describe definable sets in structures, recognizing how this depends on the language chosen;
g) apply the compactness theorem, as well as tests for completeness;
h) Understanding of the real numbers from the point of view of set theory and model theory.

Learning outcomes
To present both informal and axiomatic set theory as a foundation for mathematics. To develop the theory of ordinals and cardinals including arithmetical operations, and to introduce some basic number systems via set theory. To convey the notions of first-order structures, and of interpretations of a formula in a structure. To describe the compactness theorems of first order logic, and some of its consequences. To introduce basic notions associated with complete theories. To consider applications of both set theory and model theory. In-depth study of the real numbers from both the set-theoretic and model-theoretic point of view.

### Syllabus

We will start with a naive approach to set theory, giving the basic definitions and results around well-orderings, ordinals, cardinals, transfinite induction, and the axiom of choice. This includes ordinal and cardinal arithmetic (which is an arithmetic of “infinite” numbers in various senses). We will then cover:
- first order languages, structures, and theories;
- the compactness theorem of first order logic;
- the Lowenheim Skolem theorems, elementary equivalence, and complete theories;
- set theory as a first order theory.

We will in addition study the real numbers from both the set-theoretic and model-theoretic point of view, including nonstandard models and infinitesimals. It will also discuss informally some advanced topics such as the generalized continuum hypothesis, independence theorems in set theory, and uncountable categoricity in model theory.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Written, assessed work throughout the semester with feedback to students.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated