Math 89 Winter 2016
Set Theory

Schedule of Topics

Week 1
Monday Jan 4th Overview: context and motivations (some metamathematics)
Notes on notation
§ 1.1 , 1.2
Wednesday Jan 6th Set Arithmetic: unions, intersections, differences, products (finite and indexed) § 1.4 and pg. 27
Friday Jan 8th Paradoxes: Richard's, Russell's, Burali-Forti: linear orders and ordinals overview
First Order Logic: well-formed statements, arity, language symbols, theories, models
See Hodges § 1.1 , 1.3 for a simultaneously whimsical and formal treatment of elementary notions in model theory
Week 2
Monday Jan 11th First Order Logic/Model Theory continued: linear orders, groups, fields etc. and a survey of rudimentary results
Wednesday Jan 13th The axioms of ZFC "What sort of aggregate is verifiably a set?"
Existence of basic set arithmetic in ZFC
Justifying defined notation
§ 1.3, pg. 41, 112, 139
Suppes pg.14-18
Friday Jan 15th Relations on Sets: binary relations, orderings, functions, higher arity relations etc. Ch. 2 Hrbacek and Jech
Week 3
Monday Jan 18th No class
MLK Jr. Day
Wednesday Jan 20th Equinumerosity (Cardinality):
Injection, surjection, bijection
Finite vs. Infinite
§ 4.1 , 4.2 , 4.3 , 3.1
Thursday (X-hour) Jan 21st Equinumerosity continued:
Cantor-Bernstein (The Onion Theorem)
Faithfully representing arithmetic:
natural numbers as finite "ordinals"
§ 3.2
Friday Jan 22nd The Omega Recursion Theorem ("Finite" Recursion) § 3.3 , 4.4 , 4.6
Week 4
Monday Jan 25th Peano Arithmetic
The Ordinals!
§ 3.4 , 6.1 , 6.2
Wednesday Jan 27th The Transfinite Recursion (meta)Theorem
Ordinals can be as large as you want
§ 6.3 , 6.4
Friday Jan 29th Ordinal Arithmetic § 6.5
Week 5
Monday Feb 1st Cantor Normal Form § 6.6
Wednesday Feb 3rd The Cardinals!
von Neumann cardinal assignment
The Hartog (successor cardinal)
Aleph hierarchy vs. Beth hierarchy (CH and GCH)
Friday Feb 5th Cardinal Arithmetic:
For infinite cardinals, addition and multiplication is trivial
(The Gödel Ordering)
Week 6
Monday Feb 8th Generalized cardinal sums and products
König's Theorem
Wednesday Feb 10th Cofinality
Friday Feb 12th Cardinal exponentiation part 1
Week 7
Monday Feb 15th Cardinal exponentiation part 2:
Characterization using GCH
Wednesday Feb 17th Club and Stationary sets:
Intersecting less than cofinally many clubs
Friday Feb 19th Club and Stationary sets:
The diagonal intersection
(Fodor's) "Pressing Down" Lemma
Week 8
Monday Feb 22nd Partial ordered sets:
Chains, incomparable/incompatible elements
Maximal/minimal, greatest/least
Zorn's Lemma
Filters, subset lattice, ultrafilters
Wednesday Feb 24th The club filter and the non-stationary ideal for regular, uncountable κ
Dense sets in a poset, a brief glance at Martin's Axiom
Solutions to selected exam questions
Friday Feb 26th The well-founded universe
Relations on classes and an even more general recursion theorem
Transitive models, relations that behave like ∈,
and the Mostowski collapse function
Week 9
Monday Feb 29th Absoluteness, relativization
Reflection Theorem
Downward Löwenheim-Skolem
The countable transitive model M
Wednesday March 2nd The generic filter
The generic extension M[G] and its minimality
The forcing language
Friday March 4th non-absoluteness of cardinality
The role of the c.c.c. condition in forcing
Forcing over Fn(κ x ω, 2) and the consistency of ZFC + ¬CH
Week 10
Monday March 7th Presentations
Final Exam Assigned
March 8th-15th
The final exam will be administered and due within this time frame. Details TBA.