Set Theory

• Current Course Notes (B.Sc.) (Copyright of the author). A version of the same notes but without line numbers.

From the Course Info. Sheet:

• Live online sessions will be recorded for later repeated viewing, but note it may take up to 24 hours for these to be available. There are no face to face sessions for this course.

• All lectures and problems classes are available on the Youtube Channel to which you can subscribe (the playlist ST Set Theory is this course not the course AST Axiomatic Set Theory). There are also links on Blackboard.

• ### Problems Classes

Problem Class: live online, Tuesday 9.00, in Weeks 2-10.

• N.B., all live online sessions will be recorded for later repeated viewing, but note it may take up to 24 hours for these to be available.
• ### Maths Cafe

This will be run by Charlotte Quant cq17571@bristol.ac.uk In Weeks 2-12, Fridays 11.00-12.00. This will be via Blackboard using the Collaborate interface tool. Please send Charlotte questions in advance for these sessions.

### Lectures

Three lectures per week in Weeks 1-10: Online, live: Monday 9.00 and Tuesday 13.00; plus a pre-recorded lecture to view (either 50 minutes, or split into two parts A and B) for suggested viewing on Wednesday.

Weeks 11-12: Revision classes (also all live online).

• Breakdown of Lectures ("Lecture n" can be abbreviated "STn")
• Lecture 1 Introduction; Section 1.1
• Lecture 2 1.2
• Lecture 3 1.3
• Lecture 4 1.3.2 Ordered Pairs, Relations, Functions
• Lecture 5 1.4 Transitive Sets
• Lecture 6 2.1 Number systems: Inductive sets, Principle of Math.'l Induction
• Lecture 7 2.3 The Recursion Theorem: statement and applications [N.B. Omit Section 2.2]
• Lecture 8 2.3 The Recursion Theorem: Proof [N.B. Omit Thm. 2.18, and Example 2.19: Lines 617-635]
• Lecture 9 3 Wellorderings and ordinals pp25-26.
• Lecture 10 3 p26 + 3.1 Ordinal Numbers: Classification Theorem for Ordinals
• Lecture 11 3.1 Representation Thm. for WO's; Classification Theorem for WO's; Sect 3.2
• Lecture 12 3.2 Properties of Ordinals, Burali-Forti Lemma, sums of orderings
• Lecture 13 3.2 Products of Orderings, suprema of sets of ordinals, Recursion Theorem on On
• Lecture 14 3.2 Proof of Recursion Theorem on On, Ordinal Arithmetic
• Lecture 15 3.2 Properties of Ordinal Arithmetic
• Lecture 16 3.2 Further properties of Ordinal Arithmetic [NB Omit Lemma 3.42, Ex. 3.18]
• Lecture 17 4.1 Cardinality: Equinumerosity
• Lecture 18 4.1 Cantor-Schröder-Bernstein Theorem
• Lecture 19 4.1 Countability, WP the Wellordering Principle
• Lecture 20 4.2, 4.3 Cardinal Numbers, cardinal arithmetic
• Lecture 21 4.3 Cardinal arithmetic, Hessenberg's Theorem
• Lecture 22 4.3 Cardinal Arithmetic, cardinal exponentiation
• Lecture 23 4.3 Cardinal Arithmetic, the Generalized Continuum Hypothesis
• Lecture 24 4.3 Cardinal Arithmetic, the Generalized Continuum Hypothesis [N.B. Omit Note on Dedekind Finite Sets p.53]
• Lecture 25 5.1 Axiom of Replacement
• Lecture 26 5.2 Axiom of Choice
• Lecture 27 5.2 Axiom of Choice and its equivalents
• Lecture 28 6 Axiom of Foundation and the V-hierarchy
• Lecture 29 6 Ranks
• Lecture 30 6 Epsilon Induction
• The live on-line slots during Weeks 11 and 12 will be Revision/Problems Classes. Thus Mondays at 9.00, Tuesdays at 9.00 and 15.00. If there are sufficient queries either about Theory or Exercises/problems, I'll record a session to answer those queries for viewing later in each of these Weeks 11 and (after the break) Week 12.
Please tell me how you would like this to run, and send in your queries.

### Homework

This is set on the Monday of the Week number, and is due 12.00pm the following Monday. Please upload to Blackboard by this time.
(The system is automated so the deadline for the marker is strict.) NB The homework is not part of the course assessment (they are voluntary) but is nevertheless a necessary part of the learning process: it is there for you to practise, and you would be wise to at least attempt them each week to enhance your understanding. The Homework questions are embedded in the course text - there are no separate sheets for this. The solutions are published accordingly gradually (see 'Exercise Solutions' below).

#### Homework Sets

• Week 1 Exs. 1.2, 1.4, 1.11.
• Week 2 Exs. 1.12, 1.15, 1.26. (i) (if you find the last one easy, try part (ii)).
• Week 3 Exs. 2.3, 2.6 (a) & (f), 2.12(i).
• Week 4 Exs. 3.4, 3.6, 3.7.
• Week 5 Exs. 3.9, 3.10 (Do just N x Z isom. Z x Z ?), 3.13(ii)
• Week 6 Exs. 3.23(i), 3.27, 4.11(i).
• Week 7 Exs. 4.15(ii), 4.17 part (iii),4.20.
• Week 8 Exs. 4.34 (i) & (iii), 4.37, 4.38,
• Week 9 Exs. 5.1, 5.3, 5.7
• Week 10 Exs. 6.1,(i) and (iii); 6.2, 6.6 (use the result from Ex 6.5 for this last one) Set Monday 7th Dec.

• #### Exercise Solutions

Selected, combined for the course. NB not all Exercises are given solutions. This does not maean they are not part the course. These solution sets will only become live only after completion of the relevant homework exercises. There are no Homework sheets: the questions are all embedded in the text. There are no separate Homework Solution sheets. N.B. These solutions sets are by Chapter not by Week, and each will grow in length until the deadline for which all homework relevant to the chapter is passed.
• Reading List See the Course Information Sheet.

• Revision Topics

For past examination papers: see Blackboard ( --> Student Resources --> Examinations --> Past Examinations).

A single sample examination question and answer.