Logic, Models and Sets - 2019 entry

The goal of this module is to apply the power of mathematical reasoning to itself, and to properly examine the primary tool of Mathematics, namely logic. This study is motivated by three things. First, its own intrinsic interest. Secondly, its applications to other branches of Mathematics. We won't be able to get to any of the direct applications of logic in one semester, but this course will develop your proof skills along with other indirect applications. Finally, what logic can tell us about the nature of Mathematics. Many philosophical questions (such as, "Could all mathematics be done by a computer?''), cannot be properly answered without understanding this material.

This module will start by introducing propositional logic and deduction in propositional logic, as well as examining some of its properties (such as completeness and soundness). Then, we will do the same for predicate calculus, and then go on to discuss some of the basics of Model Theory, showing how predicate calculus can describe mathematical structures. We will then examine the Zermelo-Fraenkel axioms of Set Theory, and study the theory of ordinals and cardinals.

This module aims to give you a foundation in the study of logic, and to use that to provide an introduction to two of the branches of Logic, namely Model Theory and Set Theory. Through this course, you will become fluent in formal logical reasoning, be able to formalise mathematical structures as first order models, and clearly and rigorously answer questions about infinity.

You will be expected to have taken the Pre-requisite module, ECM2711 Groups, Rings and Fields, in the second year, but a perfect recollection will not be necessary.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Perform formal proofs in propositional logic;

2 Formalise mathematical structures into first order logic;

3 Describe the properties of a first order theory;

4 Perform cardinal and ordinal transfinite arithmetic;

5 Prove statements about any of the systems mentioned in 1-4;

Discipline Specific Skills and Knowledge:

6 State and apply definitions in the field of logic and applications;

7 Write informal proofs and translate these into formal arguments;

Personal and Key Transferable/Employment Skills and Knowledge:

8 Reason logically and abstractly;

9 Communicate complicated ideas in writing, professionally and using correct mathematical notation.

- Syntax of propositional logic. Deduction in propositional logic. Properties of propositional logic.

- Syntax of predicate calculus. Definition of model. Properties of predicate calculus.

- Definition of models, theories and axiomatisation. Definition of decidability and completeness. The Compactness Theorem.

- Introduction of ZFC. Cardinal and Ordinal arithmetic.

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE: http://vle.exeter.ac.uk/

Reading list for this module: