| Home | Homework | Readings |
| Week 1 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| Monday Jan 18th | No class MLK Jr. Day |
|
| Wednesday Jan 20th | Equinumerosity (Cardinality): Injection, surjection, bijection Finite vs. Infinite Diagonalizing |
§ 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 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| Monday Feb 8th | Generalized cardinal sums and products König's Theorem |
|
| Wednesday Feb 10th | Cofinality | |
| Friday Feb 12th | Cardinal exponentiation part 1 | |
| Week 7 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| 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 | ||
|---|---|---|
| Day | Topics | Sections |
| Monday Feb 29th | Absoluteness, relativization Reflection Theorem Downward Löwenheim-Skolem The countable transitive model M |
|
| Wednesday March 2nd | The generic filter Names 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 | ||
|---|---|---|
| Day | Topics | Sections |
| Monday March 7th | Presentations | |
| Final Exam Assigned March 8th-15th |
The final exam will be administered and due within this time frame. Details TBA. | |