## Module descriptions

### 7AAN2087 Set Theory

THIS MODULE IS RUNNING IN 2017-18

Credits: 20
Module tutor: TBC
Assessment:

2017-18

• Summative assessment: one two-hour exam (100%)
• Formative assessment: regular logic exercises

2015-16

• Summative assessment: one two-hour exam (100%)
• Formative assessment: regular logic exercises

Students are reassessed in the failed elements of assessment and by the same methods as the first attempt.

Teaching pattern: one weekly two-hour lecture and one-hour back-up class.
Additional information: the two-hour lecture will be shared with students taking 6AANA030 Set Theory, but they will otherwise be subject to different requirements
Pre-requisites:

• Some background in the study of basic formal logic is a pre-requisite for this module (eg 4AANA003 Elementary Logic or equivalent)

Sample syllabus: Please see the Past syllabi section below for an indication of the syllabus for this module.

Set Theory is important for the philosophy of mathematics and metaphysics. In this course we look at naive set theory and the paradoxes that were found in it by Russell and others; these paradoxes led to a reconstruction of the foundations of mathematics on axiomatic grounds. We will look at one such axiomatisation of set theory - Zermelo-Fraenkel set theory. Within this theory we can develop fundamental mathematical concepts that are widely used in philosophy and logic, such as the ideas of relations, partial orders, functions, partitions and equivalence classes. We also look at how natural numbers and other mathematically useful objects can be represented using sets, and we then develop a theory of the infinite using transfinite ordinal and cardinal numbers.

This course is mathematical in nature and does require some familiarity with mathematical and formal reasoning.

#### Further information

Module aims

To introduce students to:

• The class paradoxes, which revived philosophy of maths
• The theory of the transfinite
• Representing the natural numbers in the realm of sets, a gateway to foundations of maths
• Transferrable tools: Omni-usable concepts, eg function, partial/total ordering, equivalence relation
• Advanced tools: recursive definition, mathematical induction
Learning outcomes

By the end of the module, the students will be able to demonstrate intellectual, transferable and practicable skills appropriate to a Level 7 module and in particular will be able to demonstrate:

1. an understanding of the basic notion of mathematical set theory
2. a knowlege of fundamental paradoxes of naïve set theory
3. an understanding of the various kinds of infinities
4. an ability to make use of the set-theoretic construction of number systems
5. an understanding of the various mathematical orderings
Past syllabi

7AAN2087 module syllabuS 2012-13 (pdf)
7AAN2087 module syllabus 2013-14 (pdf)
7AAN2087 module syllabus 2015-16 (pdf)

Please note that module syllabus and topics covered may vary from year to year.

More detailed information on the current year’s module (including the syllabus for that year) can be accessed on KEATS by all students and staff.