# A supplement to "The Mathematics of Logic"

[NOTE: These web pages are very much under construction. Much more is planned in the next 4 months or so, and some pages that are here need review and corrections or improvement. Please contact me if you have any special requests. Richard Kaye, October 2009.]

## Contents

### Answers to selected exercises in the book

In due course I hope to have web pages containing answers or hints to all the exercises in the book. If the one you are looking for is not here yet please be patient.

### Additional exercises to test your knowledge and understanding

The following exercises test your knowledge and understanding of the marerial in the book further. They may be suitable for assessments for undergraduate courses, etc. Answers will not normally be provided on these web pages. They may not follow the order of the book exactly.

### Supplementary material on propositional and first order logic

This lists some supplementary material for propositional and first order logic, directly related to the book and available here as additional reading, some of it advanced. In particular this includes the proof of the Soundness Theorem, which is quite technical, especially when done properly. Just like the simpler examples of Soundness in the book, it is by induction on the length of proof. This sequence of web pages takes the reader through the material. Along the way you will find precise definitions of truth in an L-structure M and the precise definitions of substitution and the rules for first order logic. It is suitable for readers who want the full details and who have mastered most of The Mathematics of Logic.

### The Gödel incompleteness theorems

Possibly the most celebrated results in logic, the incompleteness theorems show there are intrinsic limitations to the idea of mechanised proof. (In other words, mathematicians are not and never will be redundant!) The pages here sketch the details and the links with computability.

### Axiomatic set theory

Axiomatic set theory is a first order theory into which all normal mathematics embeds. It formalises many arguments presented in The Mathematics of Logic including results on Zorn's Lemma and cardinal numbers.

### Some model theory

These pages build on Chapters 10 and 11 of The Mathematics of Logic. The goal is to give more examples and motivate the ideas of independence behind Morley's theorem.

• Vector spaces. Some properties of Vector Spaces over a fixed field as examples of a theory and its models.
• Embeddings. An exercise sheet on embeddings of structures, including the preservation theorem for embeddings.
• The Ryll-Nardzewski theorem. An exercise sheet applying the omitting types theorem to give a characterisation of aleph-zero categoricity.
• The theory of a successor function. Including an introduction to quantifier elimination.