Module manager: Pantelis Eleftheriou
Email: p.eleftheriou@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
MATH2140 Introduction to Logic, or equivalent
| MATH2140 | Introduction to Logic |
MATH5120M Advanced Models and Sets, MATH3120 Models and Sets
This module is not approved as an Elective
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.
To present both informal and axiomatic set theory as a foundation for mathematics. To introduce ordinals, and develop the theory of cardinals including arithmetical operations. 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.
Subject specific learning outcomes:
On successful completion of the module students will be able to:
1. Test various abstraction terms for sethood;
2. Use set theory to set up a foundation for mathematics, including constructions of some basic number systems;
3. Handle elementary arguments involving ordinals and cardinals;
4. Understand the axiom of choice;
5. Describe the relationships between first-order languages and structures, and understand the proof of the compactness theorem of first order logic;
6. Describe definable sets in structures, recognizing how this depends on the language chosen;
7. Apply the compactness theorem, as well as tests for completeness;
8. Understanding of the real numbers from the point of view of set theory and model theory.
Skills learning outcomes:
Creative problem solving
Teamwork and collaboration
Critical thinking
Rigorous writing
1. Basic set theory
2. Natural numbers and well-orders
3. Well-orders and ordinal numbers
4. Cardinal numbers
5. Languages, structures and truth
6. Expressibility, definability and elementary equivalence
7. First-order theories and compactness
8. Löwenheim–Skolem
9. Set theory as a first-order theory
10. Nonstandard analysis
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
117
Formative feedback will be provided on regular example sets or other similar learning activity.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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