Axiomatic Set Theory
Axiomatic Set Theory
Current Course Notes (M.Sci, MA) (Copyright of the author).
A version of the
same notes but without line numbers.
All lectures and problems classes are available on the
Youtube Channel to which you can subscribe,
(the playlist AST Axiomatic Set Theory is this course
not the course ST Set Theory). There are also
links on Blackboard.
Problem Class: live online, Friday 15.00, in Weeks 1-10. (The first problems class will be concerned with background
on Logic and Set Theory for those without the Bristol prerequisites (or for revision - please send me your requests
for topics to be covered here).
Live online sessions will be recorded for later repeated viewing, but note it may take up to 24 hours for
these to be available.
Review of concepts from Set Theory Please read! Note the concepts from Logic that will be assumed (at the end),
in particular the Löwenheim-Skolem Theorem.
Three lectures per week in Weeks 1-10: Monday a pre-recorded lecture to view (either 50 minutes, or
split into two parts A and B); Wednesday 10.00 & Thursday 14.00 live online lectures. Weeks 11-12:
Revision/Problems classes (also live online).
Breakdown of Lectures ("Lecture n" can be abbreviated "ASTn")
- Lecture 1 Section 1.1: Overview of the course aims; 1.2: The formal language of set theory, introduction of terms.
- Lecture 2 Section 1.2: The ZF axioms; 1.3: Transfinite Recursion.
- Lecture 3 Section 1.4: Relativisation of terms; transitive class terms.
- Lecture 4 Final part of Section 1.4; Section 2.1 Cofinality and singular ordinals.
- Lecture 5 Sect 2.1.1 More on cofinality
- Lecture 6 Sect 2.1.2 Normal Functions, Closed and Unbounded sets
- Lecture 7 Fixed points, diagonal intersections
- Lecture 8 Sect 2.1.3 Stationary Sets [NB OMIT Def. 2.17, Ex. 2.11, so lines 642-659. Also OMIT lines 690-715]
- Lecture 9 Sect 2.3 Transitive Models: the Mostowski-Shepherdson Collapsing Lemma [NB OMIT SECTION 2.2: non-examinable]
- Lecture 10 Sect 2.4 Proof of the Mostowski-Shepherdson Collapsing Lemma
- Lecture 11 Sect 2.4 The H_kappa Sets
- Lecture 12 Sect 2.4 (contd) HC, HF. 2.5 Reflection Theorems; 2.5.1 Tarski-Vaught criterion
- Lecture 13 Sect 2.5.2 Montague-Levy Reflection Theorem
- Lecture 14 Sect Further discussion on Reflection; Sect 2.6 Inaccessible cardinals
- Lecture 15 Sect 2.6.1 Inaccessible cardinals [NB OMIT Sect 2.6.2]
- Lecture 16 Sect 2.6.1 Mahlo cardinals; Sect 3.1 Definite terms and formulae
- Lecture 17 Sect 3.1 contd.) a.d. = absolutely definite properties
- Lecture 18 Sect 3.1 contd.) That wellorder is a.d., Sect 3.2 Formalising Syntax: Gödel Coding
- Lecture 19 Sect 3.2 contd.) Sect 3.3 Formalising the Satisfaction relation
- Lecture 20 Sect 3.4 Formalising definability, Sect 4.1 Definition of L-hierarchy
- Lecture 21 Sect 4.1 Properties of the L-hierarchy
- Lecture 22 Sect 4.1 Proof of ZF in L
- Lecture 23 Sect 4.2 Axiom of Choice in L
- Lecture 24 Sect 4.3 Axiom of Constructibility
- Lecture 25 Sect 4.4 The Condensation Lemma
- Lecture 26 Sect 4.4 Generalised Continuum Hypothesis in L
- Lecture 27 Sect 3.5 Incompleteness
- Lecture 28 Sect 4.5 Ordinal definability
- Lecture 29 Sect 4.5 Global Wellorders; Hereditarily Ordinal definability
- Lecture 30 Sect 4.5 Hereditarily Ordinal definability: ZFC in HOD
- The live on-line slots during Weeks 11 and 12 will be Revision/Problems Classes. Thus Wednesdays at 10.00, Thursdays 14.00 and Fridays 15.00.
If there are sufficient queries either about Theory or Exercises/problems, I'll record a session to answer those queries for viewing
at the beginning of 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.
This will now be set on the Friday of the Week number, and is due 15.00pm the following Friday. 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.
You should certainly study the solutions.
- Week 1 Exs. 1.2(i), 1.7, 1.9(iv) [you can assume you have already shown 1.9 (i)-(iii); in general if you asked to do part of a Q.
this may rely on the earlier parts, which you can assume done.]
- Week 2 Exs. 2.2, 2.6, 2.7(ii) [in the latter of you find an arbitrary
gamma < Omega difficult, just try it with gamma = omega.]
- Week 3 Exs. 2.7 (ii) (Carried over from last week, as we had not arrived at it), Ex. 2.14, 2.15
- Week 4 Exs. 2.31, 2.32, 2.33 (Officially set on Fridays for uploading following Friday by noon)
- Week 5 Exs. 2.43, 2.44, 2.47 (Set Fri. 6 Nov.)
- Week 6 Exs. 2.48 (you may find this a little tricky), 3.1, 3.2
- Week 7 Exs. 3.4, 3.5, 4.1 (set on Nov. 20)
- Week 8 Exs. 4.3, 4.5, 4.6 (set Nov. 27)
- Week 9 Exs. 4.7, 4.9, 4.12 (set Dec. 4)
- Week 10 Exs. 4.18, 4.19, 4.21 (set Dec. 11)
selected - combined for the course - these will accumulate as the course progresses.
For past examination papers: see Blackboard (--> Student Resources --> Examinations --> Past Examinations).
sample examination question and answer.